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Hadamard states for the Klein-Gordon equation onLorentzian manifolds of bounded geometry

Christian Gérard, Omar Oulghazi, Michal Wrochna

To cite this version:Christian Gérard, Omar Oulghazi, Michal Wrochna. Hadamard states for the Klein-Gordon equationon Lorentzian manifolds of bounded geometry. Annales de l’Institut Henri Poincaré, 2017, 18 (8),pp.2715-2756. 10.1007/s00023-017-0573-2. hal-01265079v3

https://hal.archives-ouvertes.fr/hal-01265079v3

https://hal.archives-ouvertes.fr

Hadamard states for the Klein-Gordon equationon Lorentzian manifolds of bounded geometry

Christian Gérard, Omar Oulghazi, and Michał Wrochna

Abstract. We consider the Klein-Gordon equation on a class of Lorentzianmanifolds with Cauchy surface of bounded geometry, which is shown to includeexamples such as exterior Kerr, Kerr-de Sitter spacetime and the maximalglobally hyperbolic extension of the Kerr outer region. In this setup, we givean approximate diagonalization and a microlocal decomposition of the Cauchyevolution using a time-dependent version of the pseudodifferential calculus onRiemannian manifolds of bounded geometry. We apply this result to constructall pure regular Hadamard states (and associated Feynman inverses), whereregular refers to the state’s two-point function having Cauchy data given bypseudodifferential operators. This allows us to conclude that there is a one-parameter family of elliptic pseudodifferential operators that encodes both thechoice of (pure, regular) Hadamard state and the underlying spacetime metric.

1. Introduction & summary of results

1.1. Introduction. Modern formulations of quantum field theory on curved space-times allow for a precise distinction between local, model-independent features, andglobal aspects specific to the concrete physical setup. In the case of non-interactingscalar fields, the study of the latter is directly related to the propagation of sin-gularities for the Klein-Gordon equation, as well as to specific global properties ofits solutions, such as two-point function positivity . Thus, a careful implementationof methods from microlocal analysis that takes into account asymptotic propertiesof the spacetime is essential in the rigorous construction of quantum fields. Thepresent paper is aimed at generalizing known methods, in particular [GW1], byproviding the necessary tools to work on a much wider class of backgrounds thatincludes examples such as Kerr and Kerr-de Sitter spacetimes.

Before formulating the problem in more detail, let us first recall how various no-tions from quantum field theory are related to inverses of the Klein-Gordon operatorand to special classes of bi-solutions.

1.1.1. Klein-Gordon equation. Consider a Klein-Gordon operator

P = −∇a∇a + V (x)

on a Lorentzian manifold (M, g), where V : M → R is a smooth function. Assumingglobal hyperbolicity1 of (M, g), the operator P has two essential properties, theproofs of which date back to Leray [Le, C-B, BGP].

The first one is the existence of retarded/advanced inverses of P , i.e. operatorsGret/adv, mapping C∞0 (M) into C∞(M) such that

P Gadv/ret = Gret/adv P = 1l, suppGret/advu ⊂ J±(suppu),

2010 Mathematics Subject Classification. 81T20, 35S05, 35L05, 58J40, 53C50.Key words and phrases. Hadamard states, pseudodifferential calculus, manifolds of bounded

geometry, Feynman parametrices.1See Subsect. 3.2.

1

Hadamard states on Lorentzian manifolds of bounded geometry 2

where J±(K) is the future/past causal shadow2 of a set K ⊂M . The second is theunique solvability of the Cauchy problem: if Σtt∈R is a foliation ofM by space-likeCauchy hypersurfaces, and ρ(t) : C∞(M) 3 φ 7→ (φΣt , i

−1∂nφΣt) ∈ C∞(Σ;C2) isthe Cauchy data operator on Σt, then, for any fixed s, there exists a unique solutionof the Cauchy problem

(1.1)

Pφ = 0,

ρ(s)φ = f

for any given f ∈ C∞0 (Σ;C2) (and moreover, ρ(t)φ ∈ C∞0 (Σ;C2) for any t ∈ R).In the present setup the two properties are actually essentially equivalent.

These two facts are basic to the theory of quantum Klein-Gordon fields on thecurved spacetime (M, g), see e.g. [Di], which we now briefly recall (see Subsect. 7.1for more details).

1.1.2. Quantum Klein-Gordon fields. By a phase space we will mean a complexvector space equipped with a non-degenerate hermitian form. The operator G =Gret −Gadv is anti-hermitian for the natural scalar product (u|v)M =

´Muv dvolg,

which allows to equip C∞0 (M) with the hermitian form u · Qv = i(u|Gv)M . Onecan show that the kernel of G equals to PC∞0 (M), hence if V =

C∞0 (M)PC∞0 (M) , (V, Q)

is a phase space — it is in fact the fundamental structure that defines the classicalcontent of the theory.

This allows one to introduce the polynomial CCR ∗-algebra CCR(V, Q), by defi-nition generated by the identity 1l and elements called the (abstract) charged fields,which are of the form ψ([u]), ψ∗([u]) for [u] ∈ C∞0 (M)

PC∞0 (M) and are subject to therelations:

i) [u] 7→ ψ([u]) is C− anti-linear,

ii) [u] 7→ ψ∗([u]) is C− linear,

iii)

[ψ([u]), ψ([v])

]=[ψ∗([u]), ψ∗([v])

]= 0,[

ψ([u]), ψ∗([v])]

= i(u|Gv)M1l,

iv) ψ([u])∗ = ψ∗([u]).

The algebraic approach to quantum field theory provides a way to represent theabove canonical commutation relations in terms of closed operators on some con-crete Hilbert space. The standard way to obtain such a representation is to specifya state.

1.1.3. Hadamard states. A state ω on CCR(V, Q) is a positive linear functional ωon CCR(V, Q) such that ω(1l) = 1. A particularly natural class of states for linearKlein-Gordon fields is the class of quasi-free states (see e.g. [DG, Sec. 17.1] andreferences therein), which are entirely determined by the expectation values:

(1.2) ω(ψ([u])ψ∗([v])

)=·· (u|Λ+v)M , ω

(ψ∗([v])ψ([u])

)=·· (u|Λ−v)M .

This definition implies in particular that P Λ± = Λ± P = 0. It is also natural torequire that Λ± : C∞0 (M)→ C∞(M), in which case Λ± have distributional kernelsΛ±(x, x′) ∈ D′(M ×M), called the two-point functions of the state ω.

Among all quasi-free states, Hadamard states are considered as the physicallyacceptable ones, because their short distance behavior resembles that of the vac-uum state on Minkowski spacetime [KW]. Since the work of Radzikowski [Ra],Hadamard states are characterized by a condition on the wave front set of their two-point functions Λ±, see Def. 7.3 for the precise statement. The use of wave front sets

2The future/past causal shadow of K ⊂M is the set of points reached from K by future/pastdirected causal (i.e. non-spacelike) curves.


had a deep impact on quantum field theory on curved spacetimes, for example onthe perturbative construction of interacting models; see e.g. [BF2, Da2, HW, KM]and also [BDH, Da1, Da3] for some recent related mathematical developments.

The microlocal formulation of the Hadamard condition in [Ra] is intimatelylinked to the notion of distinguished parametrices introduced by Duistermaat andHörmander in their influential paper [DH]. Distinguished parametrices are para-metrices of P (inverses modulo smoothing errors), which are determined uniquely(modulo smoothing errors) by the wave front set of their Schwartz kernel. Duis-termaat and Hörmander demonstrated that there are exactly four classes of dis-tinguished parametrices, the advanced/retarded and Feynman/anti-Feynman ones,see Subsect. 6.6 for details. The (uniquely defined) retarded/advanced inversesGret/adv are examples of retarded/advanced parametrices.

In contrast, there is no canonical choice of a Feynman/anti-Feynman inverse on ageneric spacetime. This is actually very closely related to the problem of specifyinga distinguished Hadamard state, see e.g. [FV2]. More specifically, the link betweenHadamard states and Feynman inverses discovered by Radzikowski is that if Λ±

are the two-point functions of a Hadamard state then the operator

GF = i−1Λ+ +Gadv = i−1Λ− +Gret

is a Feynman inverse3 of P .There already exist a large number of existence results for Hadamard states.

First of all, the deformation argument of Fulling, Narcowich and Wald [FNW] showsthat Hadamard states exist on any globally hyperbolic spacetime. This construc-tion has however the disadvantage of being very indirect, which poses problems inapplications. An alternative existence proof on arbitrary globally hyperbolic space-times was given in [GW1]. It has however another severe drawback which is thatit fails to produce pure states (see Subsect. 7.1.1) in general.

Specific examples of Hadamard states on spacetimes with special (asymptotic)symmetries include passive states for stationary spacetimes [SV], states constructedfrom data at null infinity on various classes of asymptotically flat or asymptoti-cally de Sitter spacetimes [Mo, DMP1, BJ, VW] and on cosmological spacetimes4

[DMP2, JS, BT]. Furthermore, a remarkable recent result by Sanders [Sa] provesthe existence and Hadamard property of the so-called Hartle-Hawking-Israel stateon spacetimes with a static bifurcate Killing horizon.

Finally, Junker [Ju1, Ju2] and Junker and Schrohe [JS] used the pseudodifferentialcalculus on a Cauchy hypersurface Σ to construct Hadamard states in the case ofΣ compact. The construction was then reworked in [GW1] to yield classes ofHadamard states for P in the non-compact case. Let us emphasize that outsideof the case of Σ compact, the calculus of properly supported pseudodifferentialoperators, which exists on any smooth manifold, is not sufficient to address thepositivity condition Λ± ≥ 0 and the CCR condition Λ+ − Λ− = iG which have tobe satisfied by the two-point functions Λ± in order to be consistent with (1.2) (seeSubsect. 7.1.4). This was tackled in [GW1] by assuming that the Cauchy surfaceΣ is diffeomorphic to Rd (so that the spacetime M is diffeomorphic to R1+d), withsome uniformity conditions on g at spatial infinity, which allowed to use the uniformpseudodifferential calculus on Rd.

3One could call GF a ‘time-ordered Feynman inverse’ to make the distinction with the Feynmanpropagator of Gell-Redman, Haber and Vasy [GHV, Va], which is a generalized inverse of P

considered as a Fredholm operator on suitably chosen spaces; here we just stick to the shorthandterminology.

4Let us also mention that on static and cosmological spacetimes with compact Cauchy surface,a different construction of Hadamard states was recently proposed by Brum and Fredenhagen[BF1].


In the case of spacetimes with Cauchy surfaces that are either compact or dif-feomorphic to Rd, the constructions in [Ju1, Ju2, GW1] have the advantage thatthey yield examples of Hadamard states which (in contrast to the general existenceargument from [GW1]) are pure, and it turns out that with some additional effortit is possible to obtain large classes of those. Furthermore, the two-point functionsof these states are given by rather explicit formulae (in contrast to [FNW]), fromwhich one can recover the spacetime metric, see Subsect. 1.3 for a discussion in thepresent, more general context. Unfortunately, many spacetimes of interest, like forinstance blackhole spacetimes fall outside the hypotheses in [Ju1, Ju2, GW1].

1.2. Content of the paper. In this paper we rework and extend the results of[Ju1, Ju2, GW1] in two essential directions. First of all, we greatly generalize theframework of [GW1] by basing our analysis on the pseudodifferential calculus onmanifolds of bounded geometry, due to Kordyukov [Ko] and Shubin [Sh2]. Thisallows us to work on a much larger class of spacetimes, including examples such asKerr and Kerr-de Sitter spacetimes. Secondly, the construction of Hadamard statesis now obtained as a consequence of a microlocal decomposition of the Cauchyevolution operator UA(t, s) associated to P . Beside simplifying the proofs, thisallows us to derive many formulas of independent interest, including for instanceexpressions for the Feynman inverses canonically associated to the Hadamard stateswe construct.

Let us now describe in more detail the content of the paper.The background on Riemannian manifolds of bounded geometry is presented in

Sect. 2. We use an equivalent definition of bounded geometry which is much moreconvenient in practice. In rough terms, it amounts to the existence of chart diffeo-morphisms ψxx∈M such that the pull-back metric (on Rn) (ψ−1

x )∗g is (togetherwith all derivatives) bounded above and below and equivalent to the flat metric(together with all derivatives), uniformly w.r.t. x ∈M .

This leads naturally to the notion of Lorentzian manifolds of bounded geometryand of Cauchy hypersurfaces of bounded geometry, developed in Sect. 3, which isan interesting topic in its own right. The main ingredient is the choice of a refer-ence Riemannian metric g used to define bounded tensors. We then introduce inSubsect. 3.3 a class of spacetimes and associated Klein-Gordon operators for whichparametrices for the Cauchy problem can be constructed by pseudodifferential cal-culus:

Hypothesis 1.1. We assume that there exists a neighborhood U of a Cauchy sur-face Σ in (M, g), such that:

(H) (U, g) is conformally embedded in a Lorentzian manifold of bounded geometry(M, g) and the conformal factor c2 is such that ∇g ln c is a bounded (1, 0)-tensor, moreover Σ is a so-called Cauchy hypersurface of bounded geometryin (M, g);

(M) c2V is a bounded (0, 0)-tensor.

We refer to Subsect. 3.3 for the detailed definitions. It turns out, see Sect.4, that most standard examples of spacetimes, like cosmological spacetimes, Kerr,Kerr-de Sitter, the maximal globally hyperbolic extension of Kerr, or cones, doublecones and wedges in Minkowski space belong to this class of spacetimes.

The pseudodifferential calculus on a manifold of bounded geometry is recalledin Sect. 5. A new result of importance for the analysis in the later sections of thepaper is a version of Egorov’s theorem, see Thm. 5.15.

Sect. 6 contains the main analytical results of the paper. The condition that Σis a Cauchy hypersurface of bounded geometry allows to identify the neighborhoodU with I × Σ (with I an open interval), and the Klein-Gordon equation on U can


be reduced to the standard form

(1.3) ∂2t φ+ r(t, x)∂tφ+ a(t, x, ∂x)φ = 0,

where a(t, x, ∂x) is a second order, elliptic differential operator on Σ. Denotingby UA(t, s) the Cauchy evolution operator for (1.3), mapping ρ(s)φ to ρ(t)φ, weconstruct what we call a microlocal decomposition of UA, i.e. a decomposition

(1.4) UA(t, s) = U+A (t, s) + U−A (t, s),

where U±A have the following properties, see Thm. 6.5:(1) U±A (t, s)t,s∈I are two-parameter groups (i.e. U±A (t, t′)U±A (t′, s) = U±A (t, s) for

all t, t′, s ∈ I) and U±A (t, t) =·· c±(t) are projections,(2) U±A (t, s) propagate the wave front set in the upper/lower energy shells N±,

i.e. the two respective connected components of the characteristic set of (1.3),(3) the kernels of U±A (t, s) are symplectically orthogonal for the canonical sym-

plectic form preserved by the evolution.We demonstrate in Thm. 6.8 that to such a decomposition one can associate aunique Feynman inverse for P .

Sect. 7 is devoted to the construction of Hadamard states from a microlocaldecomposition, which can be summarized as follows. We use the ‘time-kernel’notation for two-point functions Λ±, that is we write Λ±(t, s) to mean the associatedoperator-valued Schwartz kernel in the time variable. We say that a state is regularif Λ±(t, t) is a matrix of pseudodifferential operators on Σ for some t.

Theorem 1.2. Let (M, g) be a spacetime satisfying Hypothesis 1.1 and considerthe reduced Klein-Gordon equation (1.3). Let t0 ∈ I. Then there exists a pureregular Hadamard state with two-point functions given by

(1.5) Λ±(t, s) = ∓π0U±A (t, s)π∗1 ,

where π0, π1 are the projections to the respective two components of Cauchy dataand U±A (t, s)t,s∈I is a microlocal decomposition, such that

(1.6) U±A (t0, t0) =

(∓(b+ − b−)−1b∓ ±(b+ − b−)−1

∓b+(b+ − b−)−1b− ±b±(b+ − b−)−1

)(t0)

for some pair b±(t0) of elliptic first order pseudodifferential operators. Moreover,the two-point functions of any pure regular Hadamard state are of this form.

The detailed results are stated in Thm. 7.8 and 7.10, see also Prop. 7.6 forthe arguments that allow to get two-point functions for the original Klein-Gordonequation on the full spacetime (M, g) rather than for the reduced equation (1.3) onI × Σ.

Since one can get many regular states out of a given one by applying suitableBogoliubov transformations as in [GW1], Thm. 1.2 yields in fact a large class ofHadamard states.

1.3. From quantum fields to spacetime geometry. In our approach, microlo-cal splittings are obtained by setting

U±A (t, s) ··= UA(t, t0)U±A (t0, t0)UA(t0, s)

where U±A (t0, t0) is defined by formula (1.6) with b±(t) constructed for t ∈ I asapproximate solutions (i.e. modulo smoothing terms) of the operatorial equation

(1.7)(∂t + ib± + r

)(∂t − ib±

)= ∂2

t + r∂t + a,

and satisfying some additional conditions, see Sect. 6 (in particular Thm. 6.1)for details. We note that the approximate factorization (1.7) was already used byJunker in his construction of Hadamard states [Ju1, Ju2].


In summary, there is a pair of time-dependent elliptic pseudodifferential opera-tors b±(t) that uniquely determines the choice of a pure regular Hadamard state. Itis interesting to remark that b±(t) also determines the spacetime metric. First, bysubtracting the members of (1.7) one gets r(t) modulo smoothing errors. Then (1.7)gives a(t) modulo smoothing terms. But since a(t) and r(t) are differential opera-tors (the latter is just a multiplication operator), they can be determined exactly.Furthermore, the reduced operator on the r.h.s. of (1.7) is just the Klein-Gordonoperator in Gaussian normal coordinates near a Cauchy hypersurface Σt0 (see Sub-sect. 7.2), so the metric can be read in these coordinates from the knowledge of aand r.

This way, both quantum fields (derived from pure Hadamard states) and theunderlying spacetime metric are encoded by a time-dependent elliptic pseudodiffer-ential operator b+(t)⊕ b−(t). As long as one considers only pure Hadamard states(and spacetimes for which Gaussian normal coordinates make sense globally), thisprovides in particular a solution to the problem discussed in [ST] which consistsin finding a description of Hadamard states without having to specify the space-time metric explicitly. It would be thus interesting to try to build a theory whereb+(t) ⊕ b−(t) is treated as a dynamical quantity that accounts for both quantumdegrees of freedom and spacetime geometry.

We also note that the construction does not indicate directly how to select stateswith specific symmetries (in the case when the spacetime has any), which wouldbe desirable for applications and therefore deserves further investigation (see e.g.[DD] for some recent attempts).

1.4. Notation. - if X,Y are sets and f : X → Y we write f : X∼−→ Y if f is

bijective. If X,Y are equipped with topologies, we write f : X → Y if the map iscontinuous, and f : X

∼−→ Y if it is a homeomorphism.- the domain of a closed, densely defined operator a will be denoted by Dom a.- if a is a selfadjoint operator on a Hilbert space H, we write a > 0 if a ≥ 0

and Ker a = 0. We denote by 〈a〉sH the completion of Dom |a|s for the norm‖u‖s = ‖(1 + a2)s/2u‖.

- if a, b are selfadjoint operators on a Hilbert space H, we write a ∼ b if

a, b > 0, Dom a12 = Dom b

12 , c−1b ≤ a ≤ cb,

for some constant c > 0.- similarly if I ⊂ R is an open interval and Htt∈I is a family of Hilbert spaces

withHt = H as topological vector spaces, and a(t), b(t) are two selfadjoint operatorson Ht, we write a(t) ∼ b(t) if for each J b I there exist constants c1,J , c2,J > 0such that

(1.8) a(t), b(t) ≥ c1,J > 0, c2,Jb(t) ≤ a(t) ≤ c−12,Jb(t), t ∈ J.

- from now on the operator of multiplication by a function f will be denotedby f , while the operators of partial differentiation will be denoted by ∂i, so that[∂i, f ] = ∂if .

- we set 〈x〉 = (1 + x2)12 for x ∈ Rn.

2. Riemannian manifolds of bounded geometry

2.1. Definition. We recall the notion of a Riemannian manifold of bounded geom-etry, see [CG, Ro]. An important property of Riemannian manifolds of boundedgeometry is that they admit a nice ‘uniform’ pseudodifferential calculus, introducedin [Sh2, Ko], which will be recalled in Sect. 5.


2.1.1. Notation. We denote by δ the flat metric on Rn and by Bn(y, r) ⊂ Rn theopen ball of center y and radius r. If (M, g) is a Riemannian manifold and x ∈Mwe denote by BgM (x, r) (or Bg(x, r) if the underlying manifold M is clear from thecontext) the geodesic ball of center x and radius r.

We denote by rx > 0 the injectivity radius at x and by expgx : Bg(x)TxM

(0, rx)→Mthe exponential map at x.

If 0 < r < rx it is well known that expgx(Bg(x)TxM

(0, r)) = BgM (x, r) is an openneighborhood of x in M . Choosing a linear isometry ex : (Rn, δ)→ (TxM, g(x)) weobtain Riemannian normal coordinates at x using the map expgx ex.

If T is a (q, p) tensor on M , we can define the canonical norm of T (x), x ∈ M ,denoted by ‖T‖x, using appropriate tensor powers of g(x) and g−1(x). T is boundedif supx∈M ‖T‖x <∞.

Let U b Rn be open, relatively compact with smooth boundary. We denoteby C∞b (U) = C∞(Rn)U the space of smooth functions on U , bounded with allderivatives.

If V is another open set like U and χ : U → V is a diffeomorphism, we willabuse slightly notation and write that χ ∈ C∞b (U) if all components of χ belong toC∞b (U) and all components of χ−1 belong to C∞b (V ).

One defines similarly smooth (q, p) tensors on U , bounded with all derivatives.For coherence with later notation, this space will be denoted by BTpq(U, δ), whereδ is the flat metric on U . We equip BTpq(U, δ) with its Fréchet space topology.

Definition 2.1. A Riemannian manifold (M, g) is of bounded geometry if(1) the injectivity radius rg ··= infx∈M rx is strictly positive,(2) ∇kgRg is a bounded tensor for all k ∈ N, where Rg, ∇g are the Riemann

curvature tensor and covariant derivative associated to g.

We give an alternative characterization, which is often more useful in applica-tions.

Theorem 2.2. A Riemannian manifold (M, g) is of bounded geometry iff for eachx ∈M , there exists Ux open neighborhood of x and

ψx : Ux∼−→ Bn(0, 1)

a smooth diffeomorphism with ψx(x) = 0 such that if gx ··= (ψ−1x )∗g then:

(C1) the family gxx∈M is bounded in BT02(Bn(0, 1), δ),

(C2) there exists c > 0 such that:

c−1δ ≤ gx ≤ cδ, x ∈M.

A family Uxx∈M resp. ψxx∈M as above will be called a family of good chartneighborhoods, resp. good chart diffeomorphisms.

Proof. Let us first prove the ⇒ implication. We choose

Ux = expgx ex(Bn(0,r

2)) = BgM (x,

r

2),

for ex : Rn → TxM a linear isometry and ψ−1x (v) = expgx( r2exv) for v ∈ Bn(0, 1).

It is known (see e.g. [CGT, Sect. 3]) that if (M, g) is of bounded geometry, thengxx∈M is bounded in BT0

2(Bn(0, 1), δ). In fact by [Ro, Prop. 2.4] the Christof-fel symbols expressed in normal coordinates at x are uniformly bounded with allderivatives. Since ∇igjk = ∂igjk −Γlijglk = 0, this implies that all derivatives of gxin normal coordinates are bounded, hence (C1) holds. Moreover, by [Ro, Lemma2.2] we know that

mx ··=(

supX‖(expgx)∗X‖g(x) + ‖(expgx)∗X‖−1

g(x)

),


where X ranges over all unit vector fields on Bn(0, 1), is uniformly bounded inx ∈M . This is equivalent to property (C2).

Let us now prove ⇐. We first check that ∇kR is a bounded tensor for k ∈ N.Since ψx : (Ux, g)→ (Bn(0, 1), gx) is isometric, it suffices to show that

(2.1) supx∈M‖∇kgxRgx(0)‖ <∞.

In (2.1), the norm is associated to gx, but by condition (C2) we can replace it by thenorm associated to the flat metric δ. Then the l.h.s. of (2.1) is a fixed polynomialin the derivatives of gx and g−1

x computed at 0, which are uniformly bounded inx ∈M , by condition (C1). Therefore (2.1) holds.

It remains to prove that the injectivity radius rg is strictly positive. Let usdenote for a moment by r(x,N, h) the injectivity radius at x ∈ N for (N,h) aRiemannian manifold.

Clearly

(2.2) r(x, Ux, g) ≤ r(x,M, g), x ∈M.

By the isometry property of ψx recalled above, we have:

(2.3) r(x, Ux, g) = r(0, Bn(0, 1), gx).

Conditions (C1), (C2) and standard estimates on differential equations imply thatinfx∈M r(0, Bn(0, 1), gx) > 0 hence rg > 0 by (2.2), (2.3). 2

Lemma 2.3. Let Uxx∈M be the good chart neighborhoods in Thm. 2.2. Thenthere exists r > 0 such that BgM (x, r) ⊂ Ux for all x ∈M .

Proof. From condition (C2) in Thm. 2.2 we obtain the existence of some r1 > 0such that for any x ∈M Bgx(0, r1) ⊂ Bn(0, 1

2 ). Since ψx : (Ux, g)→ (Bn(0, 1), gx)is isometric, this implies that Bg(x, r1) ⊂ Ux as claimed. 2

Theorem 2.4. Let (M, g) be a manifold of bounded geometry and ε < inf(1, r, rg),where r is given in Lemma 2.3. Set

χx ··= ψx expgx ex : Bn(0, ε)→ ψx(BgM (x, ε)).

Then for any multi-index α one has:

(2.4) supx∈M,y∈Bn(0,ε)

‖Dαy χx(y)‖+ sup

x∈M,y∈ψx(BgM (x,ε))

‖Dαy χ−1x (y)‖ <∞.

Proof. Set Vx = ψx(BgM (x, ε)). Since BgM (x, ε) ⊂ Ux by Lemma 2.3, we see thatVx ⊂ Bn(0, 1). This implies (2.4) for α = 0.

Let us now consider the case |α| = 1. Since gx = (ψ−1x )∗g, we have χ∗xgx =

(expgx ex)∗g. Since (M, g) is of bounded geometry, there exists c > 0 such that

(2.5) c−1δ ≤ (expgx ex)∗g ≤ cδ.Using also condition (C2) of Thm. 2.2, we obtain c1 > 0 such that

c−11 δ ≤ gx ≤ c1δ,

hencec−11 χ∗xδ ≤ χ∗xgx ≤ c1χ∗xδ.

Since χ∗xgx = (expgx ex)∗g, we obtain:

(2.6) c−11 (expgx ex)∗g ≤ χ∗xδ ≤ c1(expgx ex)∗g.

Combining (2.5) and (2.6) we obtain c2 > 0 such that

c−12 δ ≤ χ∗xδ ≤ c2δ.

This is equivalent to (2.4) for |α| = 1.


To bound higher derivatives we use that χx is the exponential map transportedby the chart diffeomorphism ψx. Denoting by Γkij,x the Christoffel symbols for gx,we obtain that if v ∈ Bn(0, ε) ⊂ TxM and |t| ≤ 1 and (x1(t), . . . , xn(t)) ··= χx(tv)we have:

xk(t) = Γkij,x(x(t))xi(t)xj(t),

x(0) = 0,

x(0) = v.

Since Γkij,xx∈M is a bounded family in C∞b (Bn(0, 1)), it follows from standardarguments on dependence on initial conditions for differential equations that χx isuniformly bounded in C∞b (Bn(0, ε)) for x ∈M . Since we already know that Dχ−1

x

is bounded in C0(Vx), we also obtain that χ−1x is bounded in C∞b (Vx) uniformly in

x ∈M as claimed. This completes the proof of the theorem. 2

2.2. Chart coverings and partitions of unity. It is known (see [Sh2, Lemma1.2]) that if (M, g) is of bounded geometry, there exist coverings by good chartneighborhoods:

M =⋃i∈N

Ui, Ui = Uxi , xi ∈M

which are in addition uniformly finite, i.e. there exists N ∈ N such that⋂i∈I Ui = ∅

if ]I > N . Setting ψi = ψxi , we will call Ui, ψii∈N a good chart covering of M .One can associate (see [Sh2, Lemma 1.3]) to a good chart covering a partition

of unity:

1 =∑i∈N

χ2i , χi ∈ C∞0 (Ui)

such that (ψ−1i )∗χii∈N is a bounded sequence in C∞b (Bn(0, 1)). Such a partition

of unity will be called a good partition of unity.

2.3. Bounded tensors, bounded differential operators, Sobolev spaces.We now recall the definition of bounded tensors, bounded differential operatorsand of Sobolev spaces on (M, g) of bounded geometry, see [Sh2].

2.3.1. Bounded tensors.

Definition 2.5. Let (M, g) of bounded geometry. We denote by BTpq(M, g) thespaces of smooth (q, p) tensors T on M such that if Tx = (expgx ex)∗T then thefamily Txx∈M is bounded in BTpq(Bn(0, r2 ), δ). We equip BTpq(M, g) with its nat-ural Fréchet space topology.

By Thm. 2.4 we can replace in Def. 2.5 the geodesic maps expgx ex by ψ−1x ,

where ψxx∈M is any family of good chart diffeomorphisms as in Thm. 2.2.The Fréchet space topology on BTpq(M, g) is independent on the choice of the

family of good chart diffeomorphisms ψxx∈M .

2.3.2. Bounded differential operators. If m ∈ N we denote by Diffm(Bn(0, 1), δ) thespace of differential operators of orderm on Bn(0, 1) with C∞b (Bn(0, 1)) coefficients,equipped with its Fréchet space topology.

Definition 2.6. Let (M, g) of bounded geometry. We denote by Diffm(M, g) thespace of differential operators P of order m on M such that if Px = (ψ−1

x )∗P thenthe family Pxx∈M is bounded in Diffm(Bn(0, 1), δ).


2.3.3. Sobolev spaces. Let −∆g be the Laplace-Beltrami operator on (M, g), definedas the closure of its restriction to C∞0 (M).

Definition 2.7. For s ∈ R we define the Sobolev space Hs(M, g) as:

Hs(M, g) ··= 〈−∆g〉−s/2L2(M,dg),

with its natural Hilbert space topology.

It is known (see e.g. [Ko, Sect. 3.3]) that if Ui, ψii∈N is a good chart coveringand 1 =

∑i χ

2i is a subordinate good partition of unity, then an equivalent norm

on Hs(M, g) is given by:

(2.7) ‖u‖2s =∑i∈N‖(ψ−1

i )∗χiu‖2Hs(Bn(0,1)).

2.4. Embeddings of bounded geometry. We now recall the definition of em-beddings of bounded geometry, see [El].

Definition 2.8. Let (M, g) an n−dimensional Riemannian manifold of boundedgeometry, Σ an n − 1 dimensional manifold. An embedding i : Σ → M is calledof bounded geometry if there exists a family Ux, ψxx∈M of good chart diffeomor-phisms for g such that if Σx ··= ψx(i(Σ) ∩ Ux) for we have

Σx = (v′, vn) ∈ Bn(0, 1) : vn = Fx(v′),where Fxx∈M is a bounded family in C∞b (Bn−1(0, 1)).

The following fact is shown in [El, Lemma 2.27].

Lemma 2.9. Assume i : Σ → M is an embedding of bounded geometry of Σ in(M, g). Then (Σ, i∗g) is of bounded geometry.

Lemma 2.10. Let i : Σ → M an embedding of bounded geometry. Then thereexists a family Ux, ψxx∈M of good chart diffeomorphisms as in Def. 2.8 such thatif x ∈ i(Σ) one has

Σx = ψx(i(Σ) ∩ Ux) = (v′, vn) ∈ Bn(0, 1) : vn = 0.

Proof. Since the family Fxx∈Σ in Def. 2.8 is uniformly bounded in C∞0 (Bn−1(0, 1))we can find α, β > 0 such that if φx(v′, vn) = (v′, α(vn−Fx(v′))) we have Bn(0, 1) ⊂φx(Bn(0, 1)) ⊂ Bn(0, β). Clearly φxx∈Σ is a bounded family of diffeomorphismsin C∞b (Bn(0, 1)). For x ∈ Σ we replace Ux by (φxψx)−1Bn(0, 1) and ψx by φxψx.For x 6∈ Σ, Ux and ψx are left unchanged. 2

2.5. Equivalence classes of Riemannian metrics. The results of this subsec-tion are due to [Ou].

Proposition 2.11. Let (M, g) be of bounded geometry. Let k be another Riemann-ian metric on M such that k ∈ BT0

2(M, g) and k−1 ∈ BT20(M, g). Then

(1) (M,k) is of bounded geometry;(2) BTpq(M, g) = BTpq(M,k), Hs(M, g) = Hs(M,k) as topological vector spaces.Let us write k ∼ g if the above conditions are satisfied. Then ∼ is an equivalencerelation on the class of bounded geometry Riemannian metrics on M .

Proof. Let us first prove (1). We equipM with a good chart covering Ux, ψxx∈Mfor g. Then conditions (C1), (C2) of Thm. 2.2 are satisfied by k, hence (M,k) isof bounded geometry and Ux, ψxx∈M is a good chart covering for k. Using thatk ∈ BT0

2(M, g) and k−1 ∈ BT20(M, g) this implies that BTpq(M, g) = BTpq(M,k).

The statement about Sobolev spaces follows from the equivalent norm given in(2.7).


Let us show that∼ is symmetric. If g1 ∼ g2, then we have seen that BTpq(M, g1) =

BTpq(M, g2). Since (M, g2) is of bounded geometry, we have g2 ∈ BT02(M, g2) =

BT02(M, g1), g−1

2 ∈ BT20(M, g2) = BT2

0(M, g1), hence g2 ∼ g1. The same argumentshows that ∼ is transitive. 2

We conclude this subsection with an easy fact.

Proposition 2.12. Let gi, i = 1, 2 be two Riemannian metrics of bounded geometryhaving a common family of good chart diffeomorphisms Ux, ψxx∈M . Then g1 ∼g2.

Proof. This follows directly from the remark below Def. 2.5 and the definition ofthe equivalence relation g1 ∼ g2.2

2.6. Examples. We now recall some well-known examples of manifolds of boundedgeometry, which will be useful later on.

2.6.1. Compact manifolds and compact perturbations. Clearly any compact Rie-mannian manifold is of bounded geometry. Similarly if (M, g1) is of boundedgeometry and if g2 = g1 outside some compact set, then (M, g2) is of boundedgeometry and g1 ∼ g2.

2.6.2. Gluing of Riemannian manifolds. Let (Mi, gi), i = 1, 2 be two Riemannianmanifolds of bounded geometry, Ki ⊂ Mi be open and relatively compact andj : K1 → K2 an isometry. Then the Riemannian manifold (M, g) obtained bygluing M1 and M2 along K1

j∼ K2 is of bounded geometry.

2.6.3. Cartesian products. If (Mi, gi) i = 1, 2 are Riemannian manifolds of boundedgeometry then (M1 ×M2, g1 ⊕ g2) is of bounded geometry.

2.6.4. Warped products. We provide a further useful argument that gives manifoldsof bounded geometry in the form of warped products.

Proposition 2.13. Let (K,h) be a Riemannian manifold of bounded geometry,and M = Rs ×K, g = ds2 + F 2(s)h, where:(1) F (s) ≥ c0 > 0, ∀s ∈ R, for some c0 > 0;(2) |F (k)(s)| ≤ ckF (s), ∀s ∈ R, k ≥ 1.Then (M, g) is of bounded geometry.

Proof. Let r the injectivity radius of (K,h), and ey : (Rn−1, δ)→ (TyK,h(y)) fory ∈ K be linear isometries. We set for x = (σ, y) ∈M and c0 the constant in (1):

ψ−1x :

]− 1, 1[×Bn−1(0, r2c0)→]− 1, 1[×BhK(y, rc02F (σ) )

(s, v) 7→(s+ σ, exphy(F (σ)−1eyv)

).

We have:

gx = (ψ−1x )∗g = ds2 +

F 2(s+ σ)

F 2(σ)hy(eyF (σ)−1v)dv2,

where hy = (exphy ey)∗h. By (2) we have | ln(F (s+σ)F (σ) )| ≤

´ s+σσ|F′(u)F (u) |du ≤ c1|s|,

hence:e−c1 ≤ F (s+ σ)

F (σ)≤ ec1 , σ ∈ R, s ∈ ]− 1, 1[.

This implies that gx is uniformly equivalent to the flat metric δ. Moreover from con-ditions (1), (2) we obtain that gxx∈M is bounded in BT0

2(]− 1, 1[×Bn−1(0, r2c0)).We now choose ρ > 0 such that Bn(0, ρ) ⊂ ]− 1, 1[×Bn−1(0, r2c0), and compose

ψ−1x with a fixed diffeomorphism between Bn(0, ρ) and Bn(0, 1). Conditions (C1),

(C2) of Thm. 2.2 are then satisfied. 2


3. Lorentzian manifolds of bounded geometry

In this section we consider Lorentzian manifolds (M, g). Reference Riemannianmetrics on M with be denoted by g. We still denote by expgx the exponential mapat x ∈M for g. The results of this section are due to [Ou].

3.1. Definitions.

Definition 3.1. A smooth Lorentzian manifold (M, g) is of bounded geometry ifthere exists a Riemannian metric g on M such that:(1) (M, g) is of bounded geometry;(2) g ∈ BT0

2(M, g) and g−1 ∈ BT20(M, g).

Clearly the above conditions only depend on the equivalence class of g for theequivalence relation ∼ introduced in Subsect. 2.5. The following theorem is apartial converse to this property.

Theorem 3.2. Let (M, g) a Lorentzian manifold and gi, i = 1, 2 two Riemannianmetrics on M such that:(i) (M, gi) is of bounded geometry;(ii) g ∈ BT0

2(M, gi) and g−1 ∈ BT20(M, gi).

Then the following are equivalent:(1) g1 ∼ g2,(2) there exists c > 0 such that c−1g2(x) ≤ g1(x) ≤ cg2(x), ∀x ∈M ,(3) there exists c > 0 such that g2(x) ≤ cg1(x), ∀x ∈M .

Proof. We start by some preparations. Let (M, g) be a smooth Lorentzian manifoldand g a Riemannian metric on M such that (M, g) is of bounded geometry andg ∈ BT0

2(M, g), g−1 ∈ BT20(M, g). Let Ux, ψxx∈M be a family of good chart

diffeomorphisms for g and let gx = (ψ−1x )∗g.

By the above property of g and g−1, we obtain that there exists 0 < r, r′ < 1such that expgx0 is well defined on Bn(0, r), is a smooth diffeomorphism on its image,and moreover Bn(0, r′) ⊂ expgx0 Bn(0, r), and the family expgx0 x∈M is bounded inC∞b (Bn(0, r)).

Let us identify Bg(x)(0, 1) ⊂ TxM with Bn(0, 1) ⊂ Rn with isometries ex :(TxM, g(x))→ (Rn, δ) and set

φx : Bn(0, 1) 3 v 7→ expgx ex(rv) ∈M,

Vx ··= φx(Bn(0, 1)), χx = φ−1x . Since expgx0 equals expgx transported by ψx, it follows

from the properties of expgx0 x∈M shown above that Vx, χxx∈M is a family ofgood chart diffeomorphisms for g.

Let now gi, i = 1, 2 as in the theorem and let r = inf(r1, r2), where ri is the radiusr above for gi. We choose isometries ei,x : (Rn, δ) → (TxM, gi(x)) and denote byVi,x, χi,xx∈M the families of good chart diffeomorphisms for gi constructed above.

Let us compute the map Tx ··= χ1,xχ−12,x, which is defined on some neighborhood

of 0 in Bn(0, 1). Denoting by λr : Rn → Rn the multiplication by r, we have:

(3.1)χ1,x χ−1

2,x = λr (expgx e1,x)−1 (λr (expgx e2,x)−1)−1

= λr e−11,x e2,x λ−1

r = e−11,x e2,x.

We claim that property (1) is equivalent to

(3.2) supx∈M‖Tx‖+ ‖T−1

x ‖ <∞,

where ‖ · ‖ is the norm on L(Rn) inherited from δ. To prove the claim we set:

gi,x = (χ−11,x)∗gi, gi,x = (χ−1

i,x)∗gi,


so that

(3.3) g2,x = T ∗x g2,x.

We have seen above that χi,xx∈M is a family of good chart diffeomorphismsfor gi. Therefore g2,xx∈M and g−1

2,xx∈M are bounded in BT02(Bn(0, 1), δ) and

BT20(Bn(0, 1), δ). Moreover by Prop. 2.11 we know that g2 ∼ g1 iff g2,xx∈M

and g−12,xx∈M are bounded in BT0

2(Bn(0, 1), δ) and BT20(Bn(0, 1), δ). By (3.3),

this is equivalent to the fact that (T−1x )∗δx∈M and (T−1

x )∗δ−1 are boundedin BT0

2(Bn(0, 1), δ) and BT20(Bn(0, 1), δ). Since Tx are linear maps this is clearly

equivalent to (3.2), which completes the proof of the claim.Let now gi,x ··= (χ−1

i,x)∗g = (λ−1r )∗(ei,x)∗(expgx)∗g. The same argument as in

(3.1) shows that:g2,x = T ∗x g1,x = tTx g1,x Tx.

Computing the determinant of the quadratic forms gi,x(0) using δ, this implies that

(detTx)2 = det g2,x(0) det g−11,x(0).

Since gi and g−1i are bounded tensors, we obtain that there exists c > 0 such that

c−1 ≤ |detTx| ≤ c. This implies that (3.2) is equivalent to

(3.4) supx∈M‖T−1

x ‖ <∞.

Finally the discussion above shows that property (2) is equivalent to c−1g2,x ≤g2,x ≤ cg2,x ∀x ∈ M , which is equivalent to (3.2). Property (3) is equivalent tog2,x ≤ cg2,x ∀x ∈M which is equivalent to (3.4). Since we have seen that (1), (3.2)and (3.4) are equivalent, the proof is complete. 2

3.2. Cauchy hypersurfaces of bounded geometry. We adopt the conventionthat a spacetime (M, g) is a Hausdorff, paracompact, connected time orientableLorentzian manifold equipped with a time orientation. Lorentzian manifolds arenaturally endowed with a causal structure; we refer the reader to [Wa, Chap. 8] or[BGP, Sect. 1.3] for details.

In the sequel we denote by I±,gM (U), (resp. J±,gM (U)) for U ⊂M the future/pasttime-like (resp. causal) shadow of U . If (M, g) is clear from the context we useinstead the notation I±(U) (resp. J±(U)). We denote by C∞sc (M) the space ofsmooth space-compact functions, i.e. with support included in J+(K) ∪ J−(K) forsome compact set K bM .

A smooth hypersurface Σ is a Cauchy hypersurface if any inextensible piecewisesmooth time-like curve intersects Σ at one and only one point.

A spacetime having a Cauchy hypersurface is called globally hyperbolic (see [BS]for the equivalence with the alternative definition where Σ is not required to besmooth). Global hyperbolic spacetimes are natural Lorentzian manifolds on whichto study Klein-Gordon operators.

Definition 3.3. Let (M, g) be an n−dimensional Lorentzian manifold of boundedgeometry and g a Riemannian metric as in Def. 3.1. Assume also that (M, g) isglobally hyperbolic. Let Σ ⊂ M a spacelike Cauchy hypersurface. Then Σ is calleda bounded geometry Cauchy hypersurface if:(1) the injection i : Σ→M is of bounded geometry for g,(2) if n(y) for y ∈ Σ is the future directed unit normal for g to Σ one has:

supy∈Σ

n(y) · g(y)n(y) <∞.

We recall now a well-known result about geodesic normal coordinates to a Cauchyhypersurface Σ.


Proposition 3.4. Let Σ be a space-like Cauchy hypersurface in a globally hyperbolicspacetime (M, g). Then there exists a neighborhood U of 0 × Σ in R × Σ and aneighborhood V of Σ in M such that the map:

χ :U → V(s, y) 7→ expgy(sn(y))

is a diffeomorphism. Moreover χ∗gV = −ds2+hs where hs is a smooth, s−dependentfamily of Riemannian metrics on Σ.

Proof. The first statement is shown in [O’N1, Prop. 26]. To prove the secondstatement, we can work near a point in Σ and introduce local coordinates y onΣ. In [Wa, Sect. 3.3] it is shown that the normal geodesics are orthogonal to thehypersurfaces Σt = s = t. Since in the normal coordinates, ∂s is a tangent vectorto the normal geodesics, and ∂yi span TΣt this implies that the metric does notcontain dsdyi terms. If n is the future directed normal vector field to the familyΣt, then n · gn = −1 first on Σ0 and then on all Σt by the geodesic equation. Thiscompletes the proof. 2

In the next theorem we study properties of the normal coordinates for Cauchyhypersurfaces of bounded geometry.

Theorem 3.5. Let (M, g) a Lorentzian manifold of bounded geometry and Σ abounded geometry Cauchy hypersurface. Then the following holds:(1) there exists δ > 0 such that the normal geodesic flow to Σ:

χ :]− δ, δ[×Σ→M(s, y) 7→ expgy(sn(y))

is well defined and is a smooth diffeomorphism on its image;(2) χ∗g = −ds2+hs, where hss∈ ]−δ,δ[ is a smooth family of Riemannian metrics

on Σ withi) (Σ, h0) is of bounded geometry,

ii) s 7→ hs ∈ C∞b (]− δ, δ[,BT02(Σ, h0)),

iii) s 7→ h−1s ∈ C∞b (]− δ, δ[,BT2

0(Σ, h0)).

Proof. Let us first prove (1). The proof consists of several steps.Step 1: since g is of bounded geometry for the reference metric g, we first see by

standard arguments that there exists ρ2, c2 > 0 such that for all x ∈M ,

expgx : Bg(x)TxM

(0, ρ2)→M

is well defined and c2-Lipschitz if we equip Bg(x)TxM

(0, ρ2) with the distance associatedto g(x) and M with the distance associated with g.

Step 2: Recall that i : Σ → M is the natural injection. For y ∈ Σ, we setAy = D(0,y)χ ∈ L(R× TyΣ, TyM). We have:

Ay(α, v) = αn(y) +Dyiv, α ∈ R, v ∈ TyΣ,

A−1y w = (−n(y) · g(y)w, (Dyi)

−1(w + n(y) · g(y)wn(y))).

If we equip TyΣ with the metric i∗g(y) and TyM with g(y), we deduce from condi-tions (1) and (2) in Def. 3.3 that the norms of Ay and A−1

y are uniformly boundedin y.

By the local inversion theorem, there exists δ1 > 0 such that for any y ∈ Σ χ iswell defined on ]− δ1, δ1[×Bg(y, δ1) ∩ Σ and is a diffeomorphism on its image.

Step 3: let now c1 = supy∈Σ n(y) · g(y)n(y) <∞ and

δ = min(δ1, ρ2)(2 + 2c1 + 4c1c2)−1,


where ρ2, c2 are introduced at the beginning of the proof. We claim that

χ : ]− δ, δ[×Σ→M

is a smooth diffeomorphism on its image, which will complete the proof of (1). Bythe above discussion, χ is a local diffeomorphism, so it remains to prove that χ isinjective. Let (si, yi) ∈ ]− δ, δ[×Σ such that χ(s1, y1) = χ(s2, y2) = x.

If y ∈ Σ and |s| < δ, we have ‖sn(y)‖g ≤ c1δ < ρ2, hence by the Lipschitzproperty of expgx in Step 1 we have:

dg(y, expgy(sn(y))) ≤ c2|s|‖n(y)‖g.

This yields

dg(y1, y2) ≤ dg(y1, x) + dg(y2, x)

= dg(y1, expgy1(s1ny1)) + dg(y2, expgy2(s2ny2))

≤ c2|s1|‖ny1‖g + c2|s2|‖ny2‖g

≤ 2c2c1δ ≤δ12.

It follows that (si, yi) ∈ ]− δ1, δ1[×Bg(y1, δ1)∩Σ. Since by Step 2 χ is injective onthis set, we have (s1, y1) = (s2, y2), which completes the proof of (1).

Let us now prove (2). For x ∈ Σ, we choose Ux, ψx as in Lemma 2.10. Werecall that Σx = ψx(Σ ∩ Ux), gx = (ψ−1

x )∗g and denote by nx the future-directedunit normal vector field to Σx for gx. We have Σx = v ∈ Bn(0, 1) : vn = 0 ∼Bn−1(0, 1) and we can decompose nx as nx = n′x + λxen, where n′x is tangentto Σx. Then gxx∈Σ, g−1

x x∈Σ, n′xx∈Σ, λx are bounded in BT02(Bn(0, 1), δ),

BT20(Bn(0, 1), δ), BT1

0(Bn−1(0, 1), δ) and BT00(Bn−1(0, 1), δ) respectively.

By standard estimates on differential equations, this implies that there existsδ′ > 0 such that the normal geodesic flow

(3.5) χx :]− δ′, δ′[×Bn−1(0, 1

2 )→ Bn(0, 1)(s, v′) 7→ expgx(v′,0)(snx(v′, 0))

is a diffeomorphism on its image, with χxx∈Σ bounded in C∞b (]−δ′, δ′[×Bn−1(0, 12 )).

Moreover if Vx ··= χx(]− δ′, δ′[×Bn−1(0, 12 )), then χ−1

x is the restriction to Vx of amap φx : Bn(0, 1)→ Rn such that φxx∈Σ is bounded in C∞b (Bn(0, 1)).

We have χ∗xgx = −ds2 + hx(s, v′)dv′2, where hx(s, v′)dv′2 is an s−dependentRiemannian metric on Bn−1(0, 1).

To prove statement (2) it remains to check that hxx∈Σ and h−1x x∈Σ are

bounded in BT02(]−δ′, δ′[×Bn−1(0, 1

2 )) and BT20(]−δ′, δ′[×Bn−1(0, 1

2 )) respectively.This follows from the same properties of gx, g−1

x and χx recalled above. The proofis complete. 2

Remark 3.6. Since the diffeomorphisms χx in (3.5) are bounded with all deriva-tives (in good coordinates for the reference Riemannian metric g), we see that χ∗gis equivalent to ds2 + h0dy

2on I ×Σ, or more precisely that one can extend χ∗g toR× Σ such that the extension is equivalent to ds2 + h0dy

2 on R× Σ.

3.3. A framework for Klein-Gordon operators. In Sects. 6, 7 we will considerKlein-Gordon operators on a globally hyperbolic spacetime (M, g):

(3.6) P = −∇a∇a + V, V ∈ C∞(M ;R),

and in particular the Cauchy problem on a Cauchy hypersurface Σ. In this subsec-tion we formulate a rather general framework which will allow us later on to apply


tools from the pseudodifferential calculus on manifolds of bounded geometry, seeSect. 5 for the construction of parametrices for the Cauchy problem for P .

If (Mi, gi) are two spacetimes, a spacetime embedding i : (M1, g1)→ (M2, g2) isby definition an embedding that is isometric and preserves the time-orientation. Inaddition, if (Mi, gi) globally hyperbolic, one says that i is causally compatible if:

I±,g1M1(U) = i−1(I±,g2M2

(U)), ∀U ⊂M1.

We fix a globally hyperbolic spacetime (M, g), a Cauchy hypersurface Σ and afunction V ∈ C∞(M ;R). We assume that there exist:(1) a neighborhood U of Σ in M ,(2) a Lorentzian metric g on M ,(3) a function c ∈ C∞(M ;R), c > 0,such that:

(H1) (M, c2g) is globally hyperbolic, i : (U, g)→ (M, c2g) is causally compatible,(H2) g is of bounded geometry for some reference Riemannian metric g, Σ is a

Cauchy hypersurface of bounded geometry in (M, g),(H3) d ln c belongs to BT0

1(M, g),(M) c2V belongs to BT0

0(M, g).

Proposition 3.7. Assume hypotheses (H). Then there exist:(1) an open interval I with 0 ∈ I, a diffeomorphism χ : I × Σ→ U ,(2) a smooth family htt∈I of Riemannian metrics on Σ with

(Σ, h0) is of bounded geometry,

I 3 t 7→ ht ∈ C∞b (I; BT02(Σ, h0)), I 3 t 7→ h−1

t ∈ C∞b (I; BT20(Σ, h0)),

(3) a function c ∈ C∞(I × Σ), c > 0 with

∇h0ln c ∈ C∞b (I; BT1

0(Σ, h0)), ∂t ln c ∈ C∞b (I; BT00(Σ, h0)),

or equivalentlydc ∈ BT0

1(I × Σ, dt2 + h0),

such that

(3.7) χ∗g = c2(t, y)(−dt2 + ht(y)dy2) on U.

If moreover hypothesis (M) holds then:

(3.8) c2V χ−1 ∈ C∞b (I; BT00(Σ, h0)).

Proof. We apply Thm. 3.5 to g to obtain I, U, χ. We set c = c χ, so that (3.7)follows from g = c2g on U . Property (2) of t 7→ ht follow from Thm. 3.5, property(3) of c from hypothesis (H3) and the fact that χ∗g is equivalent to dt2 + h0dy

2 onI × Σ, by Remark 3.6. Finally (3.8) follows from hypothesis (M). 2

The following proposition is a converse to Prop. 3.7.

Proposition 3.8. Let (M, g) be a globally hyperbolic spacetime with

M = Rt × Σy, g = −dt2 + ht(y)dy2,

such that Σ is a Cauchy hypersurface in (M, g). Let c ∈ C∞(M), c > 0 andW ∈ C∞(M ;R). Assume that conditions (2), (3) and identity (3.7) in Prop. 3.7are satisfied by htt∈I , c for some bounded open interval I and χ = Id.

Then for any J b I conditions (H1), (H2), (H3) are satisfied for g = c2g, c = c

and U = J × Σ . If moreover V ∈ C∞(M ;R) is such that (3.8) is satisfied forχ = Id, then there exist V ∈ C∞(M ;R) such that V = V on J ×Σ and V satisfiescondition (M).


Proof. We extend the maps t 7→ ht and t 7→ c(t, ·) from I to R, in such a waythat conditions (2) and (3)are satisfied with I replaced by R, taking ht = h0,c(t, ·) = c(0, ·) for |t| large. As reference Riemannian metric on M we take g =dt2 +ht(y)dy2. The fact that (M, g) is of bounded geometry is easy. The remainingconditions in (H2), (H3) follow immediately from (2) and (3). If (3.8) holds, wecan similarly construct V with V = V on I × Σ, V = 0 for |t| large such that Vsatisfies (M). 2

4. Examples

In this section we give several examples of spacetimes to which the frameworkof Subsect. 3.3 applies.

4.1. Cosmological spacetimes. Let (Σ, h) a Riemannian manifold, a ∈ C∞(R;R)and consider M = Rt × Σy with metric

g = −dt2 + a2(t)hij(y)dyidyj .

If (Σ, h) is of bounded geometry, (M, g) satisfies conditions (H) for Σ = t = 0,c = 1, U = I × Σ, I b R an interval. Condition (M) is satisfied in particular forV = m2, m ∈ R.

Remark 4.1. The construction of propagators and Hadamard states for Klein-Gordon equations on cosmological spacetimes can be done without the pseudodif-ferential calculus used in Sects. 6, 7 in the general case. Instead one can rely onthe functional calculus for ε = (−∆h)

12 . All objects constructed in Sects. 6, 7,

like the propagators U±A (t, s) (see Subsect. 6.5) or the covariances λ±(t) (see Thm.7.8) can be written as functions of (t, s) and of the selfadjoint operator ε. Thisamounts to what is known in the physics literature as the mode decomposition, seee.g. [JS, Ol, BT, Av] for related results.

4.2. Kerr and Kerr-de Sitter exterior spacetimes.

4.2.1. The Kerr-de Sitter family. Let us recall the family of Kerr-de Sitter metrics.One setsM = Rt×Ir×S2

θ,ϕ, where I is some open interval and θ ∈ [0, π], ϕ ∈ R/2πZare the spherical coordinates on S2. The metric is given in the coordinates (t, r, θ, ϕ)(Boyer-Lindquist coordinates) by:

g = ρ2

(dr2

∆r+dθ2

∆θ

)+

∆θ sin2 θ

(1 + α)2ρ2

(adt2 − (r2 + a2)dϕ

)2− ∆r

(1 + α)2ρ2(dt− a sin2 θdϕ)2

=·· gttdt2 + gϕϕdϕ2 + 2gtϕdtdϕ+ grrdr

2 + gθθdθ2,

for∆r =

(1− α

a2r2)

(r2 + a2)− 2Mr,

∆θ = 1 + α cos2 θ, ρ2 = r2 + a2 cos2 θ,

σ2 = (r2 + a2)2∆θ − a2∆r sin2 θ.

Here α = Λa2

3 , M, a,Λ > 0 are respectively the mass of the blackhole, its angularmomentum and the cosmological constant. The Kerr metric corresponds to Λ = 0.

If Λ = 0 (Kerr) one assumes that |a| < M (slow Kerr) which implies that forrh = M +

√M2 − a2 one has:

rh > 0, ∆r(rh) = 0, ∆r > 0 on ]rh,+∞[


and one takes I =]rh,+∞[. If Λ 6= 0 (Kerr-de Sitter) one assumes that there existsrh < rc such that

i) rh > 0, ∆r > 0 on ]rh, rc[, ∆r(rh) = ∆r(rc) = 0,

ii) ∂r∆r(rh) > 0, ∂r∆r(rc) < 0,

iii) sup]rh,rc[∆r > sup[0,π] ∆θ,

and one takes I =]rh, rc[. The set S of parameters (a,M,Λ) such that suchrh, rc exist is open and contains the set |a| < M, Λ = 0 (slow Kerr) anda = 0, 9ΛM2 < 1 (Schwarzschild-de Sitter).

It is easy to check that if (a,M,Λ) ∈ S then there exists c > 0 such thatσ2(r, θ) ≥ c for all θ ∈ [0, π].

The part of the boundary r = rh of M is the (outer) black hole horizon, the partr = rc in the Kerr-de Sitter case is the cosmological horizon. Condition iii) meansthat the region ∆r > ∆θ where ∂

∂t is time-like is not empty; one chooses the timeorientation so that ∂

∂t is future oriented in this region. The spacetime M is usuallycalled the outer region of the Kerr or Kerr-de Sitter spacetime.

M

UΣ

+∞ or rcrh

+∞ or rcrh

Fig. 1 Kerr-de Sitter exterior region

4.2.2. Verification of conditions (H). The first step consists in expressing the metricin rotating coordinates. We have:

g = (gtt − g2tϕg−1ϕϕ)dt2 + gϕϕ(dϕ+ gtϕg

−1ϕϕdt)

2 + grrdr2 + gθθdθ

2.

We set R = gtϕg−1ϕϕ, ϕ = ϕ+ tR(r, θ). Denoting again ϕ by ϕ we obtain:

g = (gtt − g2tϕg−1ϕϕ)dt2 + gϕϕ(dϕ− t∂rRdr − t∂θRdθ)2 + grrdr

2 + gθθdθ2.

Then one introduces Regge-Wheeler coordinates on I, defining s = s(r) by

ds

dr= (1 + α)

r2 + a2

∆r.

(The integration constant is irrelevant). The spacetime M becomes Rt × Rs × S2ω

and we choose the Cauchy hypersurface:

Σ = M ∩ t = 0 ∼ Rs × S2ω.

We set now:

(4.1) c2 ··= −gtt + g2tϕg−1ϕϕ,

and write

g = c2g for g = −dt2 + ht, ht Riemannian metric on Σ,

with ht =·· h0 − 2th1 + t2h2.

Proposition 4.2. (1) (Σ, h0) is of bounded geometry;


(2) for J = [−ε, ε] and ε > 0 small enough one has

J 3 t 7→ ht ∈ C∞b (J ; BT02(Σ, h0)), J 3 t 7→ h−1

t ∈ C∞b (J ; BT20(Σ, h0)),

(3) One has∇h0

ln c ∈ BT10(Σ, h0), c ∈ BT0

0(Σ, h0).

Remark 4.3. By Prop. 3.8 we see that conditions (H) are satisfied. MoreoverV = m2 satisfies condition (M).

Some technical computations used in the proof of Prop. 4.2 are collected inSubsect. A.1, where the reader can also find the definitions of the function classesSpKdS and Sm,pK , see Def. A.4.

Proof of Prop. 4.2. A routine computation gives:

(4.2) c2 =∆r∆θρ

2

(1 + α)2σ2, gϕϕ =

sin2 θσ2

(1 + α)2ρ2, grr =

ρ2

∆r, gθθ =

ρ2

∆θ.

We set also:

F (s) ··= (1 + α)2 (r2 + a2)2

∆r, G(s, θ) ··=

σ2

(r2 + a2)2∆θ,

and

dω2 = dθ2 +1 + α cos2 θ

1 + αsin2 θdϕ2.

By Lemma A.2 dω2 is a smooth Riemannian metric on S2. From the identity inLemma A.2 we have

h0 =σ2

(r2 + a2)2∆θds2 +

(1 + α)2σ2

∆r∆θρ2(gθθdθ

2 + gϕϕdϕ2)

= G(s, θ)

(ds2 +

F (s)

∆θdω2 + F (s)w

),

for

w =

(a2

(1 + α)ρ2+

2ma2r

(1 + α)2ρ4

)(sin2 θdϕ)2 ∈ T0

2(Σ).

From Lemma A.7 v) we obtain that inf F (s) > 0 and |∂αs F (s)| ≤ CαF (s), hence ifk0 = ds2 + F (s)dω2, (Σ, k0) is of bounded geometry by Prop. 2.13.

Next we see from Lemma A.7 vi) that G,G−1 ∈ BT00(Σ, k0) since inf G > 0 and

∂αs (F (s)−12 ∂ω)βG is bounded on Σ for any (α, β) ∈ N3.

The factor in front of (sin2 θdϕ)2 in w belongs to S0KdS resp. to S−2,0

K . Thesame argument as the one used for G, using the estimates in Lemma A.7 showsthat F (s)w ∈ BT0

2(Σ, k0). This implies that h0 ∈ BT02(Σ, k0). Since w ≥ 0 we

immediately have that h−10 ∈ BT2

0(Σ, k0), i.e. h0 ∼ k0, which proves (1).To prove (2) we need to compute h1 and h2. We have:

h1 = c−2gϕϕRrdrdϕ+ c−2gϕϕRθ(sin 2θdθ)dϕ

=σ4

∆r∆θρ4Rrdr(sin

2 θdϕ) +σ4

∆r∆θρ4Rθ(sin 2θdθ)(sin2 θdϕ)

=σ4

∆θρ4(1 + α)(r2 + a2)Rrds(sin

2 θdϕ) +σ4

∆r∆θρ4Rθ(sin 2θdθ)(sin2 θdϕ)

=·· h1,sϕds(sin2 θdϕ) + h1,θϕ(sin 2θdθ)(sin2 θdϕ).


Similarly:

h2 = c−2gϕϕ(Rr)2dr2 + c−2gϕϕ(Rθ)

2(sin 2θdθ)2 + 2c−2gϕϕRrRθdr(sin 2θdθ)

=σ4

∆r∆θρ4sin2 θ(Rr)

2dr2 +σ4

∆r∆θρ4sin2 θ(Rθ)

2(sin 2θdθ)2

+ 2σ4

∆r∆θρ4sin2 θRrRθdr(sin 2θdθ)

=σ4

∆θρ4(r2 + a2)2(1 + α)2sin2 θ∆r(Rr)

2ds2 +σ4

∆r∆θρ4sin2 θ(Rθ)

2(sin 2θdθ)2

+ 2σ4

(1 + α)(r2 + a2)∆θρ4sin2 θRrRθds(sin 2θdθ)

=·· h2,ssds2 + h2,θθ(sin 2θdθ)2 + 2h2,sθds(sin 2θdθ).

We now collect the properties of the coefficients of h1, h2. From (A.1) and estimatessimilar to those in Lemma A.7 we obtain:

h1,sϕ ∈ S0KdS, resp. ∈ S

−1,0K , h1,θϕ ∈ S0

KdS, resp. ∈ S0,0K ,

h2,ss ∈ S−1KdS, resp. ∈ S

−4,−1K , h2,θθ ∈ S−1

KdS, resp. ∈ S−2,−1K ,

h2,sθ ∈ S−1KdS, resp. ∈ S

−3,−1K .

Since sin 2θdθ and sin2 θdϕ are smooth forms on S2, this implies that hi ∈ BT02(Σ, h0),

i = 1, 2. If J = [−ε, ε] for ε small enough we have hence

J 3 t 7→ ht ∈ C∞b (J,BT02(Σ, h0)), J 3 t 7→ h−1

t ∈ C∞b (J,BT20(Σ, h0)),

which proves (2).From (4.2) we obtain that c2 ∈ S−1

KdS, resp. ∈ S0,−1K . This implies (3). 2

4.3. Kerr-Kruskal spacetime. In this subsection we consider the maximal glob-ally hyperbolic extension of the outer Kerr region considered in Subsect. 4.2. Forthe sake of brevity we call it the Kerr-Kruskal extension. In the slow Kerr case(|a| < M,Λ = 0), ∆r has two roots 0 < r− < r+, (r+ was previously denoted by rh).The region r > r+ of Rt×Rr×S2

ω considered earlier is called the (Boyer-Lindquist)block I, the region r− < r < r+ is called the block II.

The construction of the Kerr-Kruskal extension of block I is as follows (see [O’N2,Chap. 2] for details): a block II is glued to the future of block I along r = r+, t > 0using Kerr-star coordinates, and a block II’, i.e. a block II with reversed timeorientation, is glued to the past of block I along r = r+, t < 0 using star-Kerrcoordinates. Then a block I’, i.e. a block I with reversed time orientation, is gluedto the past of block II and the future of block II’. The four blocks can be smoothlyglued together at r = t = 0 (the so-called crossing sphere), see [O’N2, Sect. 3.4].The time orientation of block I can be extended to a global time orientation, andit can be shown that the resulting spacetime (M ext, g) is globally hyperbolic, withΣext = t = 0 as a Cauchy hypersurface.


I

II

I’U0 UU ′

II’

Σext

+∞

+∞

r+

r+r+

r+

r−r−

r−r−

+∞

+∞

Fig. 2 Kerr-Kruskal extensionWe claim that the Kerr-Kruskal extension M ext satisfies the conditions (H). In

fact let U be a neighborhood of Σ in block I of the form |t| < ε, r > R, such thatProp. 4.2 holds on [−ε, ε], and let U ′ be its copy in block I’. We also fix a relativelycompact neighborhood U0 of the crossing sphere such that V = U ′ ∪ U0 ∪ U is aneighborhood of Σext in M ext. It is clear that the hypotheses of Prop. 3.8 aresatisfied, since they are satisfied over U and U ′, and U0 is relatively compact.

4.4. Double cones, wedges and lightcones in Minkowski. In this subsectionwe consider the Klein-Gordon operator P = −∇a∇a +m2 on double cones, wedgesand lightcones in Minkowski spacetime.

4.4.1. Double cones.

UU0Σ

Fig. 3 The double coneThe standard double cone is

M = (t, x) ∈ R1+d : |t| < 1− |x|, ds2 = −dt2 + dx2.

We follow the framework of Subsect. 3.3 with Σ = M ∩ t = 0, V = m2. We set

U = |t| < δ(1− |x|), t2 + (1− |x|)2 < δ for 0 < δ 1,

and fix a relatively compact open set U0 such that U ∪ U0 is a neighborhood of Σ,see Fig. 3. It suffices to check conditions (H) over U , since U0 is relatively compactin M . We introduce polar coordinates x = rω and set

r = 1− e−X cosT, t = e−X sinT.

We are reduced toU = ]− α, α[T× ]C,+∞[X×Sd−1

ω , Σ = T = 0,

ds2 = e−2X cos(2T )(−dT 2 + dX2 + 2 tan(2T )dTdX + (eX − cosT )2dω2

)


We take c(T,X) = e−X cos(2T ) and choose the reference Riemannian metric

g = dT 2 + dX2 + e2Xdω2,

which is of bounded geometry by Prop. 2.13. The Lorentzian metric

g = −dT 2 + dX2 + 2 tan(2T )dTdX + (eX − cosT )2dω2

is of bounded geometry for g. Clearly Σ = T = 0 is a bounded hypersurfaceof (U, g). Its normal vector field for g is ∂T , from which it follows that Σ is aCauchy surface of bounded geometry, hence (H2) is satisfied. One easily checksthat c satisfies (H3) and that (M) is satisfied for V = m2.

4.4.2. Wedges.

U∞U0 Σ

Fig. 4 The standard wedgeThe standard wedge is

M = (t, x1, x′) ∈ R1+d : |t| < x1, ds2 = −dt2 + dx2

1 + dx′2.

We take again Σ = M ∩ t = 0. We take:

U0 = |t| < δx1, t2 + x21 < 1, U∞ = |t| < δ, 2 < x1.

We check hypotheses (H) over U0 as above, replacing 1 − r by x1 and ω by x′.Hypotheses (H) over U∞ are immediate since g is the Minkowski metric. Thus, (H)is satisfied over U0 ∪ U∞. Hypothesis (M) is again satisfied for V = m2.

4.4.3. Lightcones in Minkowski.

UΣ

Fig.5 The future lightconeThe future lightcone is

M = (t, x) ∈ R1+d : t > |x|, ds2 = −dt2 + dx2.

We choose Σ = M ∩ t2 − x2 = 1, use polar coordinates x = rω and set

r = e−T shX, t = e−T chX,


so thatM = RT × RX × Sd−1

ω , Σ = T = 0,

ds2 = e−2T (−dT 2 + dX2 + sh2Xdω2).

We take U =] − δ, δ[T×RX × Sd−1ω as neighborhood of Σ. As before it suffices to

check hypotheses (H) over U ∩ |X| > 1. We take c(T,X) = e−T and choose thereference Riemannian metric

g = dT 2 + dX2 + e2|X|dω2

which is of bounded geometry by Prop. 2.13. Then g = −dT 2 + dX2 + sh2Xdω2

and hypotheses (H) are clearly satisfied, as is hypothesis (M) for V = m2.

5. Pseudodifferential calculus on manifolds of bounded geometry

5.1. Introduction. In this section we recall the uniform pseudodifferential calculuson a manifold of bounded geometry, due to Kordyukov [Ko] and Shubin [Sh1].This calculus generalizes for example the pseudodifferential calculus on a compactmanifold and the uniform pseudodifferential calculus on Rn. An important resultfor us is the generalization of Seeley’s theorem [Se], originally proved on a compactmanifold.

More precisely, if A ∈ Ψm(M) is an elliptic pseudodifferential operator of orderm ≥ 0 on M , symmetric and strictly positive on C∞0 (M), then A has a uniqueselfadjoint extension, still denoted by A, with domain Hm(M). Then Seeley’stheorem asserts that Az is a holomorphic family of pseudodifferential operators oforder mRez.

The extension of Seeley’s theorem to pseudodifferential operators on manifoldsof bounded geometry is due to [ALNV], which we will closely follow.

Another important result proved in this section is Egorov’s theorem. It is usuallyformulated as the fact that if A is a pseudodifferential operator and U a Fourierintegral operator then B = U−1AU is again a pseudodifferential operator. In ourcase we will take U = Uε(t, s) equal to the evolution group generated by a smoothtime-dependent family ε(t) of elliptic first order ΨDOs, with real principal symbol.

It will be convenient to consider also time-dependent pseudodifferential operatorsA = A(t) ∈ C∞b (I; Ψm(M)) for I ⊂ R an open interval. It turns out that theframework of [ALNV] is general enough to accommodate this extension withoutmuch additional work.

5.2. Symbol classes. In this subsection we recall well-known definitions aboutsymbol classes.

5.2.1. Symbol classes on Rn. Let U ⊂ Rn be an open set, equipped with the flatmetric δ on Rn.

we denote by Sm(T ∗U), m ∈ R, the space of a ∈ C∞(U × Rn) such that

〈ξ〉−m+|β|∂αx ∂βξ a(x, ξ)is bounded on U × Rn, ∀α, β ∈ Nn,

equipped with its canonical seminorms ‖ · ‖m,α,β .We set

S−∞(T ∗U) ··=⋂m∈R

Sm(T ∗U), S∞(T ∗U) ··=⋃m∈R

Sm(T ∗U),

with their canonical Fréchet space topologies.If m ∈ R and am−i ∈ Sm−i(T ∗U) we write

a '∑i∈N

am−i


if for each p ∈ N

(5.1) rp(a) ··= a−p∑i=0

am−i ∈ Sm−p−1(T ∗U).

It is well-known (see e.g. [Sh1, Sect. 3.3]) that if am−i ∈ Sm−i(T ∗U), there existsa ∈ Sm(T ∗U), unique modulo S−∞(T ∗U) such that a '

∑i∈N am−i.

We denote by Smh (T ∗U) ⊂ Sm(T ∗U) the space of a such that a(x, λξ) = λma(x, ξ),for x ∈ U , |ξ| ≥ C, C > 0.

We denote by Smph(T ∗U) ⊂ Sm(T ∗U) the space of a such that a '∑i∈N am−i

for a sequence am−i ∈ Sm−ih (T ∗U).Following [ALNV], we equip Smph(T ∗U) with the topology defined by the semi-

norms of am−i in Sm−i(T ∗U) and rp(a) in Sm−p−1(T ∗U), (see (5.1)). This topologyis strictly stronger than the topology induced by Sm(T ∗U).

The space Smph(T ∗U)/Sm−1ph (T ∗U) is isomorphic to Smh (T ∗U), and the image of

a under the quotient map is called the principal part of a and denoted by apr.Finally we note that if U = Bn(0, 1) (more generally if U is relatively compact

with smooth boundary), there exists a continuous extension map E : Sm(T ∗U)→Sm(T ∗Rn) such that EaT∗U= a. Moreover E maps Smph(T ∗U) into Smph(T ∗Rn) andis continuous for the topologies of Smph(T ∗U) and Smph(T ∗Rn), which means that allthe maps

a 7→ (Ea)m−i, a 7→ rp(Ea),

are continuous.

5.2.2. Time-dependent symbol classes on Rn. let I ⊂ R an open interval. We willalso need to consider time-dependent symbols a(t, x, ξ) ∈ C∞(I × T ∗U).

The space C∞b (I;Sm(T ∗U)) is naturally defined as the space of a ∈ C∞(I×T ∗U)such that

〈ξ〉−m+|β|∂γt ∂αx ∂

βξ a(x, ξ)is bounded on I × U × Rn, ∀α, β ∈ Nn, γ ∈ N,

equipped with its canonical seminorms ‖ · ‖m,α,β,γ . The notation a ∼∑i am−i

and the subspaces C∞b (I;Smph(T ∗U)) are defined accordingly, by requiring uniformestimates on I of all time derivatives.

5.2.3. Symbol classes on M .

Definition 5.1. We denote by Sm(T ∗M) the space of a ∈ C∞(T ∗M) such that foreach x ∈M , ax ··= (ψ−1

x )∗a ∈ Sm(T ∗Bn(0, 1)) and the family axx∈M is boundedin Sm(T ∗Bn(0, 1)). We equip Sm(T ∗M) with the seminorms

‖a‖m,α,β = supx∈M‖ax‖m,α,β .

Similarly we denote by Smph(T ∗M) the space of a ∈ Sm(T ∗M) such that for each x ∈M , ax ∈ Smph(T ∗Bn(0, 1)) and the family axx∈M is bounded in Smph(T ∗Bn(0, 1)).We equip Smph(T ∗M) with the seminorms

‖a‖m,i,p,α,β = supx∈M‖ax‖m,i,p,α,β .

where ‖ · ‖m,i,p,α,β are the seminorms defining the topology of Smph(T ∗Bn(0, 1)).

It is easy to see that the definition of Sm(T ∗M), Smph(T ∗M) and their Fréchetspace topologies are independent on the choice of the Ux, ψxx∈M , with the aboveproperties.

The notation a '∑i∈N am−i for am−i ∈ Sm−iph (T ∗M) is defined as before. If

a ∈ Smph(T ∗M), we denote again by apr the image of a in Smph(T ∗M)/Sm−1ph (T ∗M).


If I ⊂ R is an open interval, the spaces C∞b (I;Sm(T ∗M)) and C∞b (I;Smph(T ∗M))are defined as in 5.2.2.

5.3. Pseudodifferential operators. We now recall standard facts about the as-sociated pseudodifferential operators, see [Ko, Sh1, ALNV].

5.3.1. Pseudodifferential operators on Rn. If a ∈ Sm(T ∗Rn), we denote by Opw(a)its Weyl quantization, defined by

Opw(a)u(x) = (2π)−nˆ

ei(x−y)·ξa(x+y

2 , ξ)u(y)dydξ.

We recall the following well-known properties:(1) Opw(a) : C∞0 (Rn)→ E ′(Rn) is continuous,(2) Op : Sm(T ∗Rn)→

⋂s∈RB(Hs(Rn), Hs−m(Rn)) is continuous, where Hs(Rn)

is the Sobolev space of order s on Rn.(3) there exists a bilinear continuous map

S∞(T ∗Rn)× S∞(T ∗Rn) 3 (a, b) 7→ a]b ∈ S∞(T ∗Rn)

such that Opw(a)Opw(b) = Opw(a]b).

5.3.2. Time-dependent pseudodifferential operators on Rn. If I ⊂ R is an openinterval and a = a(t) ∈ C∞b (I;Sm(T ∗M)) we can consider the time-dependentpseudodifferential operator Opw(a(t)). We have(1) Opw(a(t)) : C∞b (I;C∞0 (Rn))→ C∞b (I; E ′(Rn)) is continuous,(2) Op : C∞b (I;Sm(T ∗Rn))→

⋂r,s∈RB(Hr(I;Hs(Rn)), Hr(I;Hs−m(Rn))) is con-

tinuous, where Hr(I;Hs(Rn)) is the Sobolev space of bi-order r, s on I ×Rn.

5.3.3. Quantization maps. We now recall the quantization procedure on a manifoldof bounded geometry. Let Ui, ψii∈N be a good chart covering of M and∑

i∈Nχ2i = 1l

a subordinate good partition of unity, see Subsect. 2.2. Let

(ψ−1i )∗dg =·· midx,

so that mii∈N is bounded in C∞b (Bn(0, 1)). We set also:

Ti : L2(Ui, dg)→ L2(Bn(0, 1), dx),

u 7→ m12i (ψ−1

i )∗u,

so that Ti : L2(Ui, dg)→ L2(Bn(0, 1), dx) is unitary.

Definition 5.2. Let a = a(t) ∈ C∞b (I;Sm(T ∗M)). We set

Op(a) ··=∑i∈N

χiT∗i Opw(Eai) Tiχi,

where ai = axi (see Def. 5.1), and E is the extension map (see Subsect. 5.2).Clearly Op(a) : C∞b (I;C∞0 (M))→ C∞b (I; E ′(M)) is continuous.

Such a map Op obtained from a good chart covering and partition of unity willbe called a good quantization map.

Note that Op(1) = 1l, and that Op(a)∗(t) = Op(a)(t) on C∞0 (M), where A∗ isthe adjoint of A for the scalar product

(u|v)M =

ˆM

uv dvolg.


Note that if A(t) ∈ C∞b (I; Op(S∞(T ∗M))), then its distributional kernel A(t, x, y)is supported in

(x, y) ∈M ×M : d(x, y) ≤ C,for some C > 0, where d is the geodesic distance on M . It follows that Op(a) :C∞0 (M) → C∞0 (M), hence Op(a) Op(b) is well defined. However because ofthe above support property Op(S∞(T ∗M)) is not stable under composition. Toobtain an algebra of operators, it is necessary to add to Op(S∞(T ∗M)) an ideal ofsmoothing operators, which we introduce below.

The Sobolev spaces Hs(M, g) defined in 2.3.3 will be simply denoted by Hs(M).We will set:

(5.2) H∞(M) =⋂m∈Z

Hm(M), H−∞(M) =⋃m∈Z

Hm(M),

equipped with their natural topologies.

Definition 5.3. We set:

W−∞(M) ··=⋂m∈N

B(H−m(M), Hm(M)),

equipped with its natural topology given by the seminorms

‖A‖m = ‖(−∆g + 1)m/2A(−∆g + 1)m/2‖B(L2(M)).

Similarly we equipC∞b (I;W−∞(M))

with the topology given by the seminorms

‖A‖m,p = supt∈I,k≤p

‖∂kt A(t)‖m.

The following result, showing the independence modulo C∞b (I;W−∞(M)) ofOp(C∞b (I;S∞(T ∗M))) of the above choices of Ui, ψi, χi, is easy to prove.

Proposition 5.4. Let Op′ another good quantization map. Then

Op−Op′ : C∞b (I;S∞(T ∗M))→ C∞b (I;W−∞(M)).

is continuous.

5.3.4. The axioms of a Weyl algebra. In [ALNV], a set of abstract axioms wasintroduced, with the aim of defining pseudodifferential operators on a manifold ina very general framework. The main result of [ALNV] is the extension of Seeley’stheorem [Se]. We will now check the abstract axioms of [ALNV, Sect. 1] in oursituation. Namely, we need to specify a tuple (∪k≥1W−∞k ,H, q, ]) that satisfies thefollowing properties (we refer the reader to [ALNV, 1.2] for the precise formulationin the general case):

Axiom (i): the LF-algebra and the Hilbert space: One requires thatH is a Hilbertspace and W−∞ = ∪k≥1W−∞k is a LF-algebra5, continuously embedded in B(H)and such that the adjoint operation ∗ mapsW−∞ →W−∞ continuously. We chooseH = L2(I;L2(M,dg)) and W−∞ = W−∞k = C∞b (I;W−∞(M)). The LF-algebraproperties are immediate. Furthermore, we have indeed C∞b (I;W−∞(M)) ⊂ B(H)and (

C∞b (I;W−∞(M)))∗

= C∞b (I;W−∞(M)).

Axiom (ii): existence of an injective, self-adjoint operator in W−∞: We choosethe (time-independent) operatorR = e−(∆g+1). ClearlyR = R∗ ∈ C∞b (I;W−∞(M)).

5This means that W−∞ is a strict inductive limit of Fréchet spaces and is endowed with analgebra structure with some additional grading and continuity properties, see [ALNV, 1.2].


Axiom (iii): quantization map q: We choose q(a) ··= Op(a). One needs to checkin our case that

Op(a) : L2(I;H−∞(M))→ L2(I;H−∞(M)),

Op : C∞b (I;S−∞(T ∗M))→ C∞b (I;W−∞(M)),

which is straightforward from the properties of Op already listed.Axiom (iv): It is easy to check using for example the norm given in (2.7) that

Op(a) ∈ C∞b (I;B(Hs(M), Hs−m(M))) for a ∈ C∞b (I;Sm(T ∗M)). This impliesthat

Op(C∞b (I;S∞(T ∗M)))C∞b (I;W−∞(M)) ⊂ C∞b (I;W−∞(M)),

which is in our setting the required property of the quantization map q.Axiom (v): existence of a symbolic calculus: from the symbolic calculus in

Opw(Sm(T ∗Rn)) we obtain the existence of a bilinear map

(a, b) 7→ a]b defined on C∞b (I;S∞(T ∗M))

such that

Op(a)Op(b)−Op(a]b) ∈ C∞b (I;W−∞(M)), for a, b ∈ C∞b (I;S∞(T ∗M)).

Concretely we havea]b =

∑i∈N

χ2iψ∗i (ai]bi),

where ai]bi is recalled at the beginning of Subsect. 5.2. The fact that a]b asan asymptotic expansion in terms of homogeneous bi-differential operators followsfrom the analogous property of the symbolic calculus on Rn.

Axiom (vi): boundedness of ΨDOs: from the analogous property on Rn we easilyobtain that

Op : C∞b (I;S0(T ∗M))→ B(L2(I;L2(M)))is continuous.

Axiom (vii): One requires that the map

C∞b (I;Sm(T ∗M))× C∞b (I;W−∞(M)) 3 (a, T ) 7→ Op(a) T ∈ C∞b (I;W−∞(M))

is continuous. This follows from axiom (vi) in our situation.

Two further important conditions are introduced in [ALNV].The first condition, called condition (σ) in [ALNV] amounts to the property

that if a ∈ C∞(I;Smph(T ∗M)) and Op(a) ∈ C∞b (I;W−∞(M)), then a belongs toC∞b (I;S−∞(T ∗M)). In our case we deduce from the properties of the ΨDO calculuson Rn that the sequence aii∈N is uniformly bounded in C∞b (I;S−∞(T ∗Bn(0, 1))),which implies that a ∈ C∞b (I;S−∞(T ∗M)).

The second condition, called condition (ψ) in [ALNV], is the spectral invarianceof the algebra 1l + C∞b (I;W−∞(M)). This condition is stated and proved in thelemma below.

Lemma 5.5. Let R−∞ ∈ C∞b (I;W−∞(M)) such that 1l − R−∞ is invertible inB(L2(I;L2(M))). Then

(1l−R−∞)−1 = 1l−R1,−∞ for R1,−∞ ∈ C∞b (I;W−∞(M)).

Proof. On L2(I;L2(M)) ∼´ ⊕IL2(M)dt we have:

1l−R−∞ =

ˆ ⊕I

1l−R−∞(t)dt,

hence

(1l−R−∞)−1 =

ˆ ⊕I

(1l−R−∞(t))−1dt,


and‖(1l−R−∞)−1‖B(L2(I;L2(M))) = ess supt∈I‖(1l−R−∞(t))−1‖B(L2(M))

= supt∈I‖(1l−R−∞(t))−1‖B(L2(M)),

sinceI 3 t 7→ (1l−R−∞(t))−1 ∈ B(L2(M))

is norm continuous. We have

(5.3) (1l−R−∞(t))−1 = 1l +R−∞(t) +R−∞(t)(1l−R−∞(t))−1R−∞(t).

Since (1l−R−∞(t))−1 ∈ B(L2(M)) andR−∞(t) ∈ W−∞(M), we see thatR−∞(t)(1l−R−∞(t))−1R−∞(t) ∈ W−∞(M). To prove that R1,−∞ ∈ C∞b (I;W−∞(M)) we dif-ferentiate (5.3) w.r.t. t using the Leibniz rule and the identity

∂t(1l−R−∞(t))−1 = (1l−R−∞(t))−1∂tR−∞(t)(1l−R−∞(t))−1. 2

5.3.5. Time-dependent pseudodifferential operators onM . We can now define classesof time-dependent pseudodifferential operators on M , by applying the abstractframework of [ALNV, Sect. 1]. We will only consider classical pseudodifferentialoperators, i.e. operators obtained from poly-homogeneous symbols.

Definition 5.6. We set for m ∈ R:C∞b (I; Ψm(M)) ··= Op(C∞b (I;Smph(T ∗M))) + C∞b (I;W−∞(M)).

Remark 5.7. An element of C∞b (I; Ψm(M)) will usually be denoted by A, whileA(t) for t ∈ I will be an element of Ψm(M). Writing for example L2(I;L2(M)) as´ ⊕IL2(M)dt, we have

A =

ˆ ⊕I

A(t)dt.

Note that C∞b (I; Ψ−∞(M)) = C∞b (I;W−∞(M)). If necessary we equip the spaceC∞b (I; Ψm(M)) with the quotient topology obtained from the map

C∞b (I;Smph(T ∗M))× C∞b (I;W−∞(M)) ∈ (a,R) 7→ Op(a) +R ∈ C∞b (I; Ψm(M)).

It follows that the injection:

C∞b (I; Ψm(M))→⋂s∈R

C∞b (I;B(Hs(M), Hs−m(M)))

is continuous.

Definition 5.8. Let A = Op(a) +R−∞ ∈ C∞b (I; Ψm(M)). We denote by σpr(A) ∈C∞b (I;Smh (T ∗M)) the principal symbol of A defined as

σpr(A) ··= [a] ∈ C∞b (I;Smph(T ∗M))/C∞b (I;Sm−1ph (T ∗M)).

By property (σ) and Prop. 5.4 σpr(A) is independent on the decomposition of A asOp(a) +R−∞ and on the choice of the good quantization map Op.

Definition 5.9. A ∈ C∞b (I; Ψm(M)) is elliptic if there exists C > 0 such that

|σpr(A)(t, x, ξ)| ≥ C(ξ · g−1(x)ξ)m/2, t ∈ I, (x, ξ) ∈ T ∗M.

The main property of elliptic operators is that they admit parametrices, i.e.inverses modulo C∞b (I;W−∞(M)).

Proposition 5.10. Let A ∈ C∞b (I; Ψm(M)) be elliptic. Then there exists B ∈C∞b (I; Ψ−m(M)), unique modulo C∞b (I;W−∞(M)) such that

AB − 1l ∈ C∞b (I;W−∞(M)), BA− 1l ∈ C∞b (I; (W−∞(M)).

Such an operator B is called a parametrix of A and denoted by A(−1).


Proof. The proof given in [Ko, Thm. 3.3] or [Sh2, Prop. 3.4] extends immediatelyto the time-dependent situation. 2

We recall that the notation a ∼ b for a, b are two selfadjoint operators on aHilbert space H is defined in Subsect. 1.4.

Proposition 5.11. Let A ∈ C∞b (I; Ψm(M)), m ≥ 0 be elliptic such that A(t) issymmetric on H∞(M) for all t ∈ I. Then(1) A(t) is essentially selfadjoint on H∞(M) and

DomAcl(t) = Hm(M).

(2) If in addition σpr(A)(t, x, ξ) ≥ c(ξ · g−1(x)ξ)m/2 for some c > 0, then Acl(t) isbounded below, uniformly for t ∈ I. Moreover there exists R ∈ C∞b (I; (W−∞(M))such that

A(t) +R−∞(t) ∼ (−∆g + 1)m/2, uniformly for t ∈ I.(3) A (considered as a linear operator on L2(I;L2(M))) is essentially selfadjoint

on L2(I;H∞(M)) and

DomAcl = L2(I;Hm(M)).

Proof. statement (1) follows from [ALNV, Prop. 2.2] and the alternative charac-terization of Sobolev spaces given in [ALNV, Sect.3]. To prove (2) we may assumethat A = Op(a) since W−∞(M) ⊂ B(L2(M)). Then

A(t) =∑i∈N

χiT∗i Ai(t)Tiχi,

where Aii∈N is a bounded family in Opw(C∞b (I;Sm(T ∗Rn))) such that

σpr(Ai)(t, x, ξ) ≥ c|ξ|m, uniformly for i ∈ N, t ∈ I.From the ΨDO calculus on Rn we deduce that Ai(t) ≥ c′1l uniformly in i ∈ N andwhich shows that A(t) is bounded below uniformly in t ∈ I. This also implies thatfor c 1 one has A(t) + c ∼ (−∆g + 1)m/2. By functional calculus we can findχ ∈ C∞0 (R) such that A(t) + χ(A(t)) ∼ A(t) + c. By elliptic regularity we knowthat χ(A) ∈ C∞b (I;W−∞(M)), which completes the proof of (2). (3) follows from(1). 2

We now state the main result of this subsection, which follows directly from[ALNV], for the simpler case of real powers.

Theorem 5.12. Let A ∈ C∞b (I; Ψm(M)) be elliptic, selfadjoint with A(t) ≥ c1lfor c > 0, t ∈ I. Then As ∈ C∞b (I; Ψms(M)) for any s ∈ R and

σpr(As)(t) = σpr(A(t))s.

Proof. We consider A as a selfadjoint operator on L2(I;L2(M)) and apply [ALNV,Thm. 8.9], noting that As(t) = A(t)s. 2

The following lemma will be used in Subsect. 7.4.

Lemma 5.13. Let A ∈ Ψ∞(M) such that A : E ′(M) → C∞(M). Then A ∈W−∞(M).

Proof. We can assume that A = Op(a) for a ∈ Smph(T ∗M), i.e (see Def. 5.2):

A =∑i∈N

χiT∗i Opw(Eai) Tiχi,

where Eaii∈N is bounded in Smph(T ∗Rn). We can fix cutoff functions χi such thatTiχi = χiTiχi, χii∈N is bounded in C∞0 (B(0, 1)) and define bi by χi Opw(Eai)


χi = Opw(bi). The family bii∈N is bounded in Smph(T ∗Rn) hence for each p ∈ None has:

bi =

p∑k=0

bi,m−k + ri,p,

where bi,m−ki∈N, resp. ri,pi∈N is bounded in Sm−kh (T ∗Rn) resp. Sm−p−1(T ∗Rn).Since A : E ′(M) → C∞(M) it follows that Opw(bi) : L2(Rn) → H−m+k(Rn) forany k ∈ N. Taking k = 1 we obtain that Opw(bi,m) : L2(Rn) → H−m+1(Rn),hence bi,m = 0 since bi,m is homogeneous of degree m. Iterating this argument weobtain that bi = ri,p hence Opw(bi)i∈N is bounded in B(Hs(Rn), Hs−m+p(Rn)).But this implies that A ∈ B(Hs(M), Hs−m+p(M)), using the characterization ofSobolev spaces in 2.3.3. Since p is arbitrary we have A ∈ W−∞(M).2

5.4. Egorov’s theorem. Let us consider an operator ε(t) = ε1(t) + ε0(t), suchthat:

(E)εi(t) ∈ C∞b (I; Ψi(M)), i = 0, 1,

ε1(t) is elliptic, symmetric and bounded from below on H∞(M).

(see Def. 5.9). By Prop. 5.11 we know that ε1(t) with domain Dom ε(t) = H1(M)is selfadjoint, hence ε(t) with the same domain is closed, with non empty resolventset. We denote by Uε(t, s) the associated propagator defined by:

∂∂tUε(t, s) = iε(t)Uε(t, s), t, s ∈ I,∂∂sUε(t, s) = −iUε(t, s)ε(s), t, s ∈ I,

Uε(s, s) = 1l, s ∈ I.Note that the propagator Uε1(t, s) exists and is unitary on L2(M), by e.g. [RS,Thm. X.70]. Since ε(t) − ε1(t) is uniformly bounded in B(L2(M)), one easilydeduces the existence of Uε(t, s), which is strongly continuous in (t, s) ∈ I2 withvalues in B(L2(M)), uniformly bounded on I2 in B(L2(M)).

Lemma 5.14. Assume (E). Then(1) Uε(t, s) ∈ B(Hm(M)) for m ∈ Z ∪ ±∞, I2 3 (t, s) 7→ Uε(t, s) is strongly

continuous on Hm(M),(2) if r−∞ ∈ W−∞(M) then Uε(t, s)r−∞, r−∞Uε(t, s) ∈ C∞b (I2

t,s,W−∞(M)).

Proof. Note that (2) follows from (1). If clearly suffices to prove (1) for m finite.We set a = (−∆g + 1l)

12 and compute

∂t(Uε(s, t)amUε(t, s)a−m

)= −iUε(s, t)[ε(t), am]Uε(t, s)a−m

= −iUε(s, t)× [ε(t), am]a−m × amUε(t, s)a−m.

We know that am ∈ Ψm(M), hence [ε(t), am]a−m ∈ C∞(I; Ψ0(M)). MoreoverUε(t, s) is locally bounded in B(L2(M)) on I2. Therefore

∂t‖Uε(s, t)amUε(t, s)a−mu‖ ≤ C‖Uε(s, t)amUε(t, s)a−mu‖, (t, s) ∈ I2, u ∈ L2(M).

For m < 0, taking u ∈ Dom a−m and using Gronwall’s inequality yields (1). Form > 0 we argue similarly, replacing the unbounded operator am by amδ = am(1l +iδa)−m for δ > 0. We obtain from Gronwall’s inequality that:

‖Uε(s, t)amδ Uε(t, s)a−m‖ ≤ C‖amδ a−m‖, (t, s) ∈ I2.

We conclude the proof by using that ‖amu‖ = sup0<δ ‖amδ u‖.2


The following theorem is a version of Egorov’s theorem.

Theorem 5.15. Let a ∈ Ψm(M) and ε(t) satisfying (E). Then

a(t, s) ··= Uε(t, s)aUε(s, t) ∈ C∞b (I2,Ψm(M)).

Moreoverσpr(a)(t, s) = σpr(a) Φ(s, t),

where Φ(t, s) : T ∗M → T ∗M is the flow of the time-dependent Hamiltonian σpr(ε)(t).

Proof. The proof consists of several steps.Step 1: we write a = Op(c) + a−∞, c ∈ Smph(T ∗M), a−∞ ∈ W−∞. Then

Uε(t, s)aUε(s, t) = Uε(t, s)Op(c)Uε(s, t) + Uε(t, s)a−∞Uε(s, t)

= Uε(t, s)Op(c)Uε(s, t) + C∞b (I2,W−∞(M)),

by Lemma 5.14. Therefore we can assume that a = Op(c).Step 2: we write ε(t) = Op(b)(t)+ε−∞(t) for b(t) ∈ C∞b (I;S1

ph(T ∗M)), ε−∞(t) ∈C∞b (I2,W−∞(M)).

We write:Uε(t, s) =·· UOp(b)(t, s)V(t, s),

where ∂tV(t, s) = −iUOp(b)(s, t)ε−∞(t)Uε(t, s) =·· ε−∞(t, s),

V(s, s) = 1l.

By Lemma 5.14 we know that ε−∞(t, s) ∈ C∞b (I2,W−∞(M)), hence

V(t, s) = 1l + C∞b (I2,W−∞(M)).

It follows that:Uε(t, s)Op(c)Uε(s, t) = UOp(b)(t, s)V(t, s)Op(c)V(s, t)UOp(b)(s, t)

= UOp(b)(t, s)Op(c)UOp(b)(s, t) + C∞b (I2,W−∞(M)),

again by Lemma 5.14. Therefore it suffices to consider

a1(t, s) ··= UOp(b)(t, s)Op(c)UOp(b)(s, t).

Step 3: We try to construct d(t, s) ∈ C∞b (I2, Smph(T ∗M)) such that

(5.4)

∂tOp(d)(t, s) = −[Op(b)(t), iOp(d)(t, s)], t, s ∈ I,

Op(d)(s, s) = Op(c), s ∈ I,

modulo error terms in W−∞(M). As in [Ta, Sec. 0.9], we write

c '∑i∈N cm−i, cm−i ∈ S

m−ih (T ∗M),

b(t) '∑i∈N b1−i(t), b1(t) = σpr(ε)(t), b1−i(t) ∈ C∞(I;S1−i

ph (T ∗M)),

and solve (5.4) with the ansatz

d(t, s) '∑i∈N

dm−i(t, s), dm−i ∈ C∞(I;Sm−ih (T ∗M)).

We obtain the sequence of transport equations:

(E0)

∂tdm(t, s) + σpr(ε(t)), dm(t, s) = 0,

dm(s, s) = cm,

(Ei)

∂tdm−i(t, s) + σpr(ε(t)), dm(t, s) =

∑−j+m−k+1−l=m−i Pj(dm−k, b1−l)(t, s),

dm−i(s, s) = 0,


where ·, · is the Poisson bracket and

Pj : Sp1h (T ∗M)× Sp2h (T ∗M)→ Sp1+p2−jh (T ∗M)

is a bi-differential operator homogeneous of degree j (see [ALNV, Sect. 1.1]).This sequence of transport equations can be solved inductively in

C∞b (I;Sm−ih (T ∗M)),

using that σpr(ε(t)) is real-valued and elliptic. We have in particular:

(5.5) dm(t, s) = cm Φ(s, t).

Now we choose d(t, s) '∑i∈N dm−i(t, s) and obtain:∂tOp(d)(t, s) = −[Op(b)(t), iOp(d)(t, s))] + C∞b (I2,W−∞(M))

Op(d)(s, s) = Op(c) + C∞b (I;W−∞(M)).

It follows that∂t(UOp(b)(s, t)Op(d)(t, s)UOp(b)(t, s)

)= UOp(b)(s, t) (∂tOp(d)(t, s)− i[Op(b)(t),Op(d)(t, s)])UOp(b)(t, s)

∈ C∞b (I2,W−∞(M)),

UOp(b)(s, s)Op(d)(s, s)UOp(b)(s, s) = Op(c) +W−∞(M).

Hence by integrating from s to t and using again Lemma 5.14:

Op(d)(t, s) = a1(t, s) + C∞b (I2,W−∞(M)).

Hence a1(t, s) ∈ C∞(I2,Ψm(M)) as claimed. By (5.5) we have:

σpr(a1(t, s)) = σpr(a) Φ(s, t).

The proof is complete. 2

5.5. The wave front set. In this subsection we recall the characterization of thewave front set of a distribution u ∈ E ′(M) using pseudodifferential operators onM .One says that A ∈ Ψm(M) is elliptic at (x0, ξ0) ∈ T ∗M\0 if

σpr(A)(x0, ξ0) 6= 0.

Proposition 5.16. Let u ∈ D′(M). Then (x0, ξ0) ∈ T ∗M\0 does not belongto WF(u) iff there exists A ∈ Ψ0(M), elliptic at (x0, ξ0) and χ ∈ C∞0 (M) withχ(x0) 6= 0 such that Aχu ∈ H∞(M), or equivalently χAχu ∈ C∞0 (M).

Let us also recall some more notation. If Mi, i = 1, 2 are two manifolds oneidentifies T ∗(M1×M2) and T ∗M1×T ∗M2. IfK : C∞0 (M2)→ D′(M1) is continuous,denoting again by K ∈ D′(M1 ×M2) its distributional kernel, one sets:

WF(K)′ ··= (X1, X2) ∈ (T ∗M1 × T ∗M2)\0 : (X1, X2) ∈WF(K),

where (x, ξ) = (x,−ξ).

Proposition 5.17. Let Uε(t, s) be as in Thm. 5.15. Then:

WF(Uε(t, s)u) = Φ(t, s)(WF(u)), u ∈ H−∞(M),

WF(Uε(t, s))′ = (X,Y ) ∈ T ∗M\0 × T ∗M\0 : X = Φ(t, s)(Y )

Proof. This follows immediately from Prop. 5.16, Thm. 5.15 and the fact thatUε(t, s) preserves H∞(M). 2


6. Parametrices and propagators

6.1. Introduction. In this section we consider a class of model Klein-Gordon equa-tions of the form

(KG) ∂2

tφ+ r(t, x)∂tφ+ a(t, x, ∂x)φ = 0

on It × Σ, where I ⊂ R is an open interval. We will see that the Klein-Gordonequations introduced in Subsect. 3.3 can be reduced to such model equations.We will consider the associated Cauchy evolution operator UA(t, s), mapping ρ(s)φ

to ρ(t)φ for ρ(t)φ ··=(

φ(t)i−1∂tφ(t)

). It is well-known (see e.g. [Ch]) that UA(t, s)

can expressed microlocally as the sum of two Fourier integral operators, associatedwith the symplectic flow Φ±(t, s) generated by ±(a2(t, x, ξ))

12 , where a2(t, x, ξ) is

the principal symbol of a(t, x, ∂x).This fact is not sufficient for our purposes, namely the construction of pure

Hadamard states for Klein-Gordon fields. We need a more precise decompositionof UA(t, s) as a sum

UA(t, s) = U+A (t, s) + U−A (t, s)

which we call amicrolocal decomposition (see Subsect. 6.5). The essential propertiesrequired of U±A (t, s) is that they are evolutions groups, propagate the wave frontset by the flows Φ±(t, s) and that their ranges are symplectically orthogonal for thenatural symplectic form preserved by UA(t, s).

On a technical level, we avoid the use of the Fourier integral operators machineryand rely instead on propagators Ub(t, s) generated by time-dependent ΨDOs, whichwere studied in Subsect. 5.4. As a by-product of the construction of U±A (t, s), wealso obtain a Feynman inverse for the operator P in (KG), canonically associatedwith the corresponding state, see Subsect. 6.6.

6.2. The model Klein-Gordon equation. In this subsection we give the preciseassumptions on our model Klein-Gordon operator (KG), to which the Klein-Gordonoperators considered in Subsect. 3.3 can be reduced.

We fix an open interval I ⊂ R with 0 ∈ I and a smooth d−dimensional manifoldΣ, equipped with a Riemannian metric k0, such that (Σ, k0) is of bounded geometry.

We fix the following objects:(1) a time-dependent Riemannian metric ht on Σ such that ht ∈ C∞b (I; BT0

2(Σ, k0))

and h−1t ∈ C∞b (I; BT2

0(Σ, k0)),(2) a differential operator a(t, x, ∂x) ∈ C∞b (I; Diff2(Σ, k0)) such that

i) σpr(a)(t, x, ξ) = ξ · h−1t (x)ξ,

ii) a(t, x, ∂x) = a∗(t, x, ∂x)

where the adjoint is defined using the time-dependent scalar product

(6.1) (u|v) =

ˆΣ

uv|ht|12 dx.

We define then the model Klein-Gordon operator:

P = ∂2

t + r(t, x)∂t + a(t, x, ∂x),

for r(t, x) ··= |ht|−12 ∂t|ht|

12 . This way P is formally selfadjoint for the scalar prod-

uct:

(u|v)M =

ˆI×Σ

uv|ht|12 dtdx.


6.3. Solutions to a Riccati equation. Let us abbreviate a(t, x, ∂x), r(t, x) sim-ply by a, r. The essential step in the construction of parametrices of the Cauchyproblem for the model Klein-Gordon equation introduced in Subsect. 6.2 is to findtime-dependent operators b±(t) ∈ C∞(I; Ψ1(Σ)) such that the associated evolutionoperators Ub±(t, s) solve(

∂2

t + r∂t + a)Ub±(t, s) = 0, modulo smoothing errors.

The above equation is equivalent to the following Riccati equation:

(6.2) i∂tb± − b±2 + a+ irb± = 0,

again modulo smoothing errors. In [GW1] (6.2) was solved in the special case Σ =Rd, r = 0, using the uniform pseudodifferential calculus on Rd, and an equivalentequation (see (6.10)) was solved before by Junker in the case of Σ compact [Ju1,Ju2]. In this subsection we extend the construction to the case when Σ is a manifoldof bounded geometry, using the pseudodifferential calculus described in Sect. 5,allowing also for r 6= 0.

Applying Prop. 5.11 to a, we can find c > 0 and c−∞ ∈ C∞b (I;W−∞(Σ))

such that a(t) + c−∞(t) ≥ c1 for t ∈ I. We set ε(t) = (a(t) + c−∞(t))12 , so that

ε2(t) = a(t) + C∞b (I;W−∞(Σ)). Since a is elliptic, we know from Thm. 5.12 thatε ∈ C∞b (R,Ψ1(Σ)), with principal symbol (ξ · h−1

t (x)ξ)12 .

Theorem 6.1. There exists b ∈ C∞b (I; Ψ1(Σ)), unique modulo C∞b (I;W−∞(Σ))such that

i) b = ε+ C∞b (I; Ψ0(Σ)),

ii) (b+ b∗)−1 = (2ε)−12 (1l + r−1)(2ε)−

12 , r−1 ∈ C∞b (I; Ψ−1(Σ)),

iii) (b+ b∗)−1 ≥ cε−1, for some c ∈ C∞b (I;R), c > 0,

iv) i∂tb± − b±2 + a+ irb± = r±−∞ ∈ C∞b (I;W−∞(Σ)),

for b+ ··= b, b− ··= −b∗.

Proof. We follow the proof in [GW1, Appendix A3]. We can first replace in(6.2) a by ε2, modulo an error term in C∞b (I;W−∞(Σ)). Discarding error termsin C∞b (I;W−∞(Σ)), we can assume that ε = Op(c), c ∈ C∞b (I;S1

ph(T ∗Σ)), withcpr(t, x, ξ) = (ξ · h−1

t (x)ξ)12 . We look for b of the form b = Op(c) + Op(d) for

d ∈ C∞b (I;S0ph(T ∗Σ)). Since Op(c) is elliptic, it admits parametrices, see Prop.

5.10. We fix a symbol c ∈ C∞b (I;S−1ph (T ∗Σ)) such that Op(c) is a parametrix of

Op(c).The equation (6.2) becomes, modulo error terms in C∞b (I;W−∞(Σ)):

(6.3) Op(d) =i

2(Op(c)Op(∂tc) + Op(c)rOp(c)) + F (Op(d)),

for:

F (Op(d)) =1

2Op(c)

(iOp(∂td) + [Op(c),Op(d)] + irOp(d)−Op(d)2

).

From symbolic calculus, we obtain that:

F (Op(d)) = Op(F (d)) + C∞b (I;W−∞(Σ)),

forF (d) =

1

2c] (i∂td+ c]d− d]c+ ir]d− d]d) ,

where the operation ] (the Moyal product) is recalled in 5.3.4. The equation (6.3)becomes:

(6.4) d = a0 + F (d),


fora0 =

i

2(c]∂tc+ c]r]c) ∈ C∞b (I;S0

ph(T ∗Σ)).

The map F has the following property:

(6.5)d1, d2 ∈ C∞b (I;S0

ph(T ∗Σ)), d1 − d2 ∈ C∞b (I;S−jph (T ∗Σ))

⇒ F (d1)− F (d2) ∈ C∞b (I;S−j−1ph (T ∗Σ)).

This allows to solve symbolically (6.4) by setting

d−1 = 0, dn ··= a0 + F (dn−1),

andd '

∑n∈N

dn − dn−1,

which is an asymptotic series since by (6.5) we see that dn−dn−1 ∈ C∞b (I;S−nph (T ∗Σ)).It follows that Op(c+ d) solves (6.2) modulo C∞b (I;W−∞(Σ)).

We observe then that if b ∈ C∞b (I; Ψ∞(Σ)) we have:

(∂tb)∗ = ∂t(b

∗) + rb∗ − b∗r,(recall that the adjoint is computed w.r.t the time-dependent scalar product (6.1)).This implies that −Op(d)∗ is also a solution of (6.2) modulo C∞b (I;W−∞(Σ)).

To complete the construction of b±, we consider

s = Op(c+ d) + Op(c+ d)∗,

which is selfadjoint, with principal symbol equal to 2(ξ ·h−1t (x)ξ)

12 . By Prop. 5.11,

there exists r−∞ ∈ C∞b (I;W−∞(Σ)) such that

(6.6) s+ r−∞ ∼ ε,where we recall that the notation ∼ is defined in (1.8). We set now:

b ··= Op(c+ d) +1

2r−∞.

Properties i) and iv) follow from the same properties of Op(c + d). Property iii)follows from (6.6) and the Kato-Heinz theorem. To prove property ii) we write

b+ b∗ = (2ε)12 (1l + r−1)(2ε)

12 ,

where r−1 ∈ C∞b (I; Ψ−1(Σ)), by Thm. 5.12. Since (1l+r−1) is boundedly invertible,we have again by Thm. 5.12

(1l + r−1)−1 = 1l + r−1, r−1 ∈ C∞b (I; Ψ−1(Σ)),

which implies ii). The proof is complete. 2

Note that by iv) one has (by subtracting the two identities)

r = i−1(b+ + b−)− (b+ − b−)−1∂t(b+ − b−)

modulo smoothing errors. Thus, the pair b± contains full information about r, andthus about a (using iv) again).

6.4. Approximate diagonalization. In this subsection we perform a diagonal-ization modulo smoothing errors of the Cauchy evolution operator UA(t, s), see6.4.1.

We extend the notation in Sect. 5 to matrix-valued symbols, operators, etc., byintroducing the sets C∞b (I; Ψm(Σ,Cnp )), n, p ∈ N etc. We will frequently omit theextra symbol Cnp when the nature of the objects is clear from the context. We alsoextend to this situation the notation Uε(t, s) when ε ∈ C∞b (I; Ψm(Σ,Cnn)).


6.4.1. KG equation as a first order system. As usual we write

(∂2t + r(t)∂t + a(t))φ(t) = 0

as a first order system:

(6.7) i−1∂tψ(t) = A(t)ψ(t), where A(t) =

(0 1la(t) ir(t)

),

by setting

ψ(t) =

(φ(t)

i−1∂tφ(t)

)=·· ρ(t)φ.

We equip L2(Σ;C2) with the time-dependent scalar product obtained from (6.1),by setting:

(f |g) ··=ˆ

Σ

(f1g1 + f0g0)|ht|12 dx.

We will use it to define adjoints of linear operators and to identify sesquilinearforms on L2(Σ;C2) with linear operators. Note that if φi are C∞ solutions withφiΣ compactly supported then

iφ1 · σφ2 = (ρ(t)φ1|qρ(t)φ2)

for:

q ··=(

0 11 0

)is independent on t. The evolution operator UA(t, s) is symplectic:

(6.8) q = U∗A(s, t)qUA(s, t), s, t ∈ I.

6.4.2. First reduction. The Riccati equation

(6.9) i∂tb± − b±2 + a+ irb± = r±−∞

implies that:

(6.10) (∂t + ib± + r) (∂t − ib±) = ∂2

t + r∂t + a− r±−∞,

which is a factorization of the Klein-Gordon operator P modulo smoothing errors.One can also deduce from (6.10) a time-dependent diagonalization of the evolutionoperator for P , which we now define. We set

ψ(t) ··=(∂t − ib−(t)

∂t − ib+(t)

)φ(t),

and obtain ψ(t) = S−1(t)ψ(t) with(6.11)

S−1(t) = i

(−b−(t) 1−b+(t) 1

), S(t) = i−1

(1 −1

b+(t) −b−(t)

)(b+(t)− b−(t))−1,

which makes sense thanks to b+(t)−b−(t) being invertible by Thm. 6.1. We obtainfrom (6.10) that

(6.12)

(∂t + ib− + r 0

0 ∂t + ib+ + r

)ψ(t) =

(∂

2

t + a+ r∂t − r−−∞∂

2

t + a+ r∂t − r+−∞

)φ(t)

=

(PφPφ

)−(r−−∞ 0r+−∞ 0

)S(t)ψ(t),

LetB(t) = B(t) + S−∞(t),


for

B(t) =

(−b− + ir 0

0 −b+ + ir

), S−∞(t) =

(r−−∞ −r−−∞r+−∞ −r+

−∞

)(b+ − b−)−1.

Then since Pφ = 0 we deduce from (6.12) that:

(∂t − iB(t))ψ(t) = 0,

hence:

(6.13) UA(t, s) = S(t)UB(t, s)S(s)−1.

We have thus a formula that relates UA(t, s) and the evolution generated by a time-dependent operator B(t) that is diagonal up to W−∞(Σ) remainders and whoseon-diagonal terms have principal symbols ±(ξ · h−1

t (x)ξ)12 .

Let us now discuss the symplectic properties of UB(t, s). Since

S(s)∗qS(s) = (b+ − b−)−1(s)

(1 00 −1

)=·· qB(s)

we obtain from (6.13), (6.8) that:

qB(t) = U∗B(s, t)qB(s)UB(s, t), s, t ∈ I.

6.4.3. Second reduction. To get rid of the (b+ − b−)−1(s) factor in qB(s) we set

UC(t, s) ··= (b+ − b−)−12 (t)UB(t, s)(b+ − b−)

12 (s).

It follows that:

(6.14) UA(t, s) = T (t)UC(t, s)T (s)−1,

for:

(6.15)T (t) ··= S(t)(b+ − b−)

12 (t) = i−1

(1 −1b+ −b−

)(b+ − b−)−

12 ,

T−1(t) = i(b+ − b−)−12

(−b− 1−b+ 1

).

Note that:

(6.16) T ∗(t)qT (t) =

(1 00 −1

)=·· q,

so that UC(t, s) is symplectic for q:

(6.17) UC(t, s)∗qUC(t, s) = q.

The generator of UC(t, s) is:

(6.18) C(t) = C(t) +R−∞(t),

for

(6.19)

C(t) ··= (b+ − b−)−12 B(t)(b+ − b−)

12 − i∂t(b

+ − b−)−12 (b+ − b−)

12

=

(−b− + r−0 0

0 −b+ + r+0

)where

r±0 = ir + [(b+ − b−)−12 , b±]− i∂t(b

+ − b−)−12 (b+ − b−)

12 ∈ C∞b (I; Ψ0(Σ)).

and

(6.20) R−∞ = −(b+ − b−)−12S−∞(b+ − b−)

12 ∈ C∞b (I;W−∞(Σ)).


Remark 6.2. Let us explain another motivation for the introduction of the mapsT (t). There are two natural topologies on the space of Cauchy data for (6.7). Thefirst is the energy space topology given by the topology of H1(Σ)⊕L2(Σ), ubiquitousin the PDE literature. The second is the charge space topology, given by the topologyof H

12 (Σ)⊕H− 1

2 (Σ), related to the quantization of the Klein-Gordon equation. Itis easy to see that S(t) is an isomorphism from L2(Σ)⊕ L2(Σ) to H1(Σ)⊕ L2(Σ),while T (t) is an isomorphism from L2(Σ)⊕ L2(Σ) to H

12 (Σ)⊕H− 1

2 (Σ).

6.4.4. Interaction picture. From (6.18) we know that the generator of UC(t, s) isdiagonal, modulo a smoothing error term. It follows from standard argumentsthat UC(t, s) is also diagonal, modulo smoothing errors. We review this argument,known as the ‘interaction picture’ in the physics literature.

Let H(t) = H0(t) + V (t) be a time-dependent Hamiltonian, U(·, ·) and U0(·, ·)the associated propagators. We fix t1 ∈ R and set:

U(t, s) =·· U0(t, t1)Vt1(t, s)U0(t1, s).

(Typically H0 does not depend on time and one sets t1 = 0). It follows that Vt1(·, ·)is an evolution group and solves∂tVt1(t, s) = iVt1(t)Vt1(t, s) for Vt1(t) = U0(t1, t)V (t)U0(t, t1),

Vt1(s, s) = 1l.

Note the following covariance property:

Vt2(t, s) = U0(t1, t2)Vt1(t, s)U0(t2, t1), t1, t2 ∈ R.

6.4.5. Parametrix for the Cauchy problem. We apply the above procedure to C =C +R−∞, fix some t1 ∈ I and set:

UC(t, s) =·· UC(t, t1)Vt1(t, s)UC(t1, s),

where Vt1(t, s) is the evolution generated by Rt1,−∞(t) = UC(t1, t)R−∞(t)UC(t, t1),i.e.

(6.21)

∂tVt1(t, s) = iRt1,−∞(t)Vt1(t, s),

Vt1(s, s) = 1l.

Note that C(t) is diagonal, with entries satisfying condition (E) in Subsect. 5.4.Therefore by Lemma 5.14 we know that Rt1,−∞(t) ∈ C∞b (I;W−∞(Σ)). For anys ∈ R, the equation (6.21) can be solved in C∞b (I2;B(Hs(Σ))) by a convergentseries. This implies easily that:

(6.22)Vt1(t, s) = 1l + C∞b (I2,Ψ−∞(Σ)),

UC(t, s) = UC(t, s) + C∞b (I2,Ψ−∞(Σ)).

We summarize this discussion with the following theorem.

Theorem 6.3. Let

(6.23) UA(t, s) ··= T (t)UC(t, s)T (s)−1.

Then UA(t, s)(t,s)∈I2 is an evolution group and:

UA(t, s) = UA(t, s) + C∞b (I2,Ψ−∞(Σ)).

It follows that the group UA(t, s)(t,s)∈I2 is a parametrix for the Cauchy problem.

Note that since C(t) is diagonal, we have:

(6.24) UC(t, s) =

(U−b−+r−0

(t, s) 0

0 U−b++r+0(t, s)

).


6.5. Decomposition of the Cauchy evolution. Basing on the constructions inSubsect. 6.4 it is easy to construct a microlocal decomposition of the evolutionUA(t, s). In fact let

(6.25) π+ =

(1l 00 0

), π− =

(0 00 1l

).

We fix a reference time t0 ∈ I for example t0 = 0 and set:

(6.26) c±(0) ··= T (0)π±T−1(0) =

(∓(b+ − b−)−1b∓ ±(b+ − b−)−1

∓b+(b+ − b−)−1b− ±b±(b+ − b−)−1

)(0).

We have:

c±(0)2 = c±(0), c+(0) + c−(0) = 1l, c±(0) ∈ C∞b (I; Ψ∞(Σ)).

It follows that c+(0), c−(0) is a pair of complementary projections. Moreover from(6.16), we obtain that:

(6.27) c∓∗(0)qc±(0) = 0,

i.e. the ranges of the projections c±(0) are q−orthogonal. We set:

(6.28) U±A (t, s) ··= UA(t, 0)c±(0)UA(0, s).

Definition 6.4. A pair U±A (t, s)(t,s)∈I2 as in (6.28) will be called a microlocaldecomposition of the evolution group UA(t, s)(t,s)∈I2 .

Theorem 6.5. The following properties are true:

i) U±A (t, s)U±A (s, t′) = U±A (t, t′),

ii) U+A (t, s) + U−A (t, s) = UA(t, s),

iii) U±A (t, s)∗qU∓A (t, s) = 0,

iv) (∂t − iA(t))U±A (t, s) = U±A (t, s)(∂s − iA(s)) = 0,

v) WF(U±A (t, s))′ = (X,X ′) ∈ T ∗Σ× T ∗Σ : X = Φ±(t, s)(X ′),

where Φ±(t, s) : T ∗Σ→ T ∗Σ is the symplectic flow generated by the time-dependentHamiltonian ±(ξ · h−1

t (x)ξ)12 .

Proof. i) and ii) follow from the fact that c±(0) are complementary projections.iii) follows from (6.8) and (6.27). iv) is immediate. From (6.24) and Prop. 5.17we obtain that UC(t, 0)π±UC(0, s) has the wave front set stated in v). The resultfollows then from the fact that U±A (t, s) = T (t)UC(t, 0)π±UC(0, s)T−1(s). 2

We now gather a couple of formulae that relate various objects at different times.The proof is a routine computation that uses the first three statements in Thm.6.5.

Proposition 6.6. Let

(6.29) c±(t) ··= U±A (t, t) = UA(t, 0)c±(0)UA(0, t).

Then:c±(t)2 = c±(t), c+(t) + c−(t) = 1l,

c±(t) = UA(t, s)c±(s)UA(s, t),

c∓(t)qc±(t)∗ = 0, c±(t)UA(t, s)c∓(s) = 0,

U±A (t, s) = c±(t)UA(t, s)c±(s) = c±(t)UA(t, s) = UA(t, s)c±(s).


6.6. The Feynman inverse associated to a microlocal decomposition. Inthis subsection we work in the setup of Subsect. 6.2

6.6.1. Distinguished parametrices for the Klein-Gordon operator. In our terminol-ogy, a continuous map G : C∞0 (M) → C∞(M) is a (two-sided) parametrix of theKlein-Gordon operator P if PG − 1l and 1l − GP have smooth kernels. In whatfollows we recall the classification of parametrices of P due to Duistermaat andHörmander in [DH].

For x ∈ M we denote by Vx± ⊂ TxM the future/past solid lightcones and byV ∗x± ⊂ T ∗xM the dual cones V ∗x± = ξ ∈ T ∗xM : ξ · v > 0, ∀v ∈ Vx±, v 6= 0. Wewrite

ξ 0 (resp. ξ 0) if ξ ∈ V ∗x+ (resp. V ∗x−).

For X = (x, ξ) ∈ T ∗M\0 denote p(X) = ξ ·g−1(x)ξ the principal symbol of P andN = p−1(0)∩ T ∗M\0 the characteristic manifold of P . If Hp is the Hamiltonianvector field of p, integral curves of Hp in N are called bicharacteristics. N splitsinto the upper/lower energy shells

N = N+ ∪N−, N± = N ∩ ±ξ 0.

For X1, X2 ∈ N we write X1 ∼ X2 if X1, X2 lie on the same bicharacteristic. ForX1 ∼ X2, we write X1 X2 (resp. X1 ≺ X2) if X1 comes strictly after (before) X2

w.r.t. the natural parameter on the bicharacteristic through X1 and X2. Finallyone sets

C = (X1, X2) ∈ N ×N : X1 ∼ X2, ∆ = (X,X) : X ∈ T ∗M\0,

andCret/adv = (X1, X2) ∈ C : x1 ∈ J±(x2),

CF = (X1, X2) ∈ C : X1 ≺ X2,

CF = (X1, X2) ∈ C : X1 X2.The main results of [DH] relevant to us is the following theorem.

Theorem 6.7. [DH, Thm. 6.5.3] For ] = ret, adv,F,F there exists a parametrixG] of P such that

(6.30) WF(G])′ = ∆ ∪ C].

Any other parametrix G with WF(G)′ ⊂ ∆∪ C] equals G] modulo a smooth kernel.

A parametrix satisfying (6.30) for ] = ret/adv resp. ] = F/F will be called aretarded/advanced resp. Feynman/anti-Feynman parametrix (or inverse if PG] = 1land G]P = 1l hold exactly).

6.6.2. The Feynman inverse associated to a microlocal decomposition. We now showhow to associate to the decomposition of the Cauchy evolution constructed in Sub-sect. 6.5 a Feynman inverse for the Klein-Gordon operator P .

In the next theorem, we will use the ‘time kernel’ notation: namely if A :C∞0 (M ;Cp) → C∞(M ;Cq) we denote by A(t, s) : C∞0 (Σ;Cp) → C∞(Σ;Cq) itsoperator-valued kernel, defined by

Au(t) =

ˆRA(t, s)u(s)ds, u ∈ C∞0 (M ;Cp).

We denote by πi : L2(Σ;C2) → L2(Σ) for i = 0, 1 the projection on the first orsecond component and by θ(s) the Heaviside function.


Theorem 6.8. Let U±A (t, s) be a microlocal decomposition and let

(6.31) GF(t, s) = i−1π0

(U+A (t, s)θ(t− s)− U−A (t, s)θ(s− t)

)π∗1 .

Then GF : C∞0 (M)→ C∞(M) is continuous and:

P GF = GF P = 1l.

Moreover WF(GF)′ = ∆ ∪ CF, hence GF is a Feynman inverse.

Proof. The fact thatGF : C∞0 (M)→ C∞(M) is continuous follows from Thm. 6.3,Lemma 5.14 and the fact that C∞0 (M) ⊂ C∞b (R;H∞(Σ)) ⊂ C∞(M) continuously.In the rest of the proof we will use freely the time-kernel notation. We will denoteby ρ the map C∞0 (M) 3 u 7→ (u, i−1∂tu) ∈ C∞0 (M ;C2), whose kernel is δ(t−s)ρ(s).

To prove that P GF = 1l, we set R(t, s) = U+A (t, s)θ(t − s) − U−A (t, s)θ(s − t).

Since (∂t − iA(t)) U±A = 0, we obtain

((∂t − iA(t)) R)(t, s) = U+A (t, s)δ(t− s) + U−A (t, s)δ(s− t)

= (c+(s) + c−(s))δ(t− s) = 1lΣδ(t− s),

hence (∂t − iA(t)) R = 1l. This implies that

π0 (∂t − iA(t)) R π∗1 = 0, π0 (∂t − iA(t)) R π∗1 = 1l,

which by an easy computation implies that P GF = 1l.To prove that GF P = 1l we note that

π∗1 P = i(∂t − iA(t)) ρ, θ(±(t− s)) ∂s = ∂s θ(±(t− s))± δ(t− s).

Using then that

U±A (∂t − iA(t)) = 0, U+A (s, s) + U−A (s, s) = c+(s) + c−(s) = 1l,

we obtain that GF P = π0 ρ = 1l. Writing X = (t, x, τ, ξ) ∈ T ∗(R × Σ)\0 wehave:

X1≺X2 ⇔ τi = ±(ξi · h−1(ti, xi)ξi)

12 , (x1, ξ1) = φ±(t1, t2)(x2, ξ2).

Using Thm. 6.5 v) this easily implies that WF(GF)′ = ∆ ∪ CF. 2

7. Hadamard states

In this section we associate to a microlocal decomposition as in Def. 6.4 a uniquepure Hadamard state ω. The Cauchy surface two-point functions (see Def. 7.4) are(matrices of) pseudodifferential operators on Σ. We give the relation between thespacetime two-point functions of ω and the operators U±A (·, ·) in Def. 6.4.

We say that a state is regular if its Cauchy surface two-point functions are (ma-trices) of pseudodifferential operators (in the sense of the calculus on manifolds ofbounded geometry). We show that any pure regular Hadamard state is actuallyassociated to a microlocal decomposition.

7.1. Klein-Gordon fields. We start by reviewing classical material about quasi-free states for Klein-Gordon fields, see e.g. [DG, KM, HW]. We use the complexformalism, based on charged (i.e., complex) fields ψ,ψ∗, which turns out to be moreconvenient for our analysis.


7.1.1. Bosonic quasi-free states. Let V be a complex vector space, V∗ its anti-dualand let us denote Lh(V,V∗) the space of hermitian sesquilinear forms on V. A pair(V, q) consisting of a complex vector space V and a non-degenerate hermitian formq on V will be called a phase space. We denote by U(V, q) the pseudo-unitary groupfor (V, q).

As outlined in the introduction, given a phase space (V, q) one can define theCCR ∗-algebra CCR(V, q) (see e.g. [DG, Sect. 8.3.1]) 6. The (complex) fieldoperators V 3 v 7→ ψ(v), ψ∗(v), which generate CCR(V, q), are anti-linear, resp.linear in v and satisfy the canonical commutation relations

[ψ(v), ψ(w)] = [ψ∗(v), ψ∗(w)] = 0, [ψ(v), ψ∗(w)] = vqw1l, v, w ∈ V.

The complex covariances Λ± ∈ Lh(V,V∗) of a state ω on CCR(V, q) are defined interms of the abstract field operators by

(7.1) v · Λ+w ··= ω(ψ(v)ψ∗(w)

), v · Λ−w ··= ω

(ψ∗(w)ψ(v)

), v, w ∈ V

Note that Λ± ≥ 0 and Λ+ − Λ− = q by the canonical commutation relations.Conversely if Λ± are Hermitian forms on V such that

(7.2) Λ+ − Λ− = q, Λ± ≥ 0,

then there is a unique quasi-free state ω such that (7.1) holds, see e.g. [DG, Sect.17.1].

In order to discuss purity of quasi-free states in terms of their two-point functions,one needs to work in a C∗-algebraic framework instead.

If VR is V considered as a real vector space and σ = i−1q, then (VR,Reσ) is areal symplectic space. We denote by W(V, q) the Weyl C∗-algebra over (VR,Reσ),see e.g. [DG, Sect. 8.5.3], whose generators are denoted by W (v). We still denoteby ω the quasi-free state on W(V, q) defined by

ω(W (v)) = e−12 vηv, for η = Re(Λ± ∓ 1

2q),

see [GW1, Sect. 2.3]. By definition ω is pure if it is pure as a state on the C∗-algebraW(V, q).

Note that (7.2) implies that Ker(Λ+ +Λ−) = 0, hence ‖v‖2ω ··= vΛ+v+vΛ−v isa Hilbert norm on V. Denoting by Vcpl the completion of V for ‖·‖ω, the hermitianforms q,Λ± extend uniquely to qcpl,Λ±,cpl on Vcpl, and ω uniquely extends to astate ωcpl on CCR(Vcpl, qcpl) or W(Vcpl, qcpl). Note that qcpl may be degenerate.

If V1 ⊂ Vcpl with V ⊂ V1 densely for ‖·‖ω, then we also obtain unique objectsq1,Λ

±1 , ω1 that extend q,Λ±, ω.

In the proposition below, we give a characterization of pure quasi-free states.Note that the characterization given in [GW1, Prop. 2.7] was incorrect, unlessV = Vcpl.

Proposition 7.1. The state ω is pure on CCR(V, q) iff there exists V1 ⊂ Vcpl withV ⊂ V1 densely for ‖·‖ω and projections c±1 ∈ L(V1) such that

(7.3) c+1 + c−1 = 1l, c+∗1 q1c−1 = 0, Λ±1 = ±q1 c±1 .

The proof is given in Appendix A.2.

6See [GW1] for the transition between real and complex vector space terminology.


7.1.2. Phase spaces for Klein-Gordon fields. Let (M, g) be a globally hyperbolicspacetime and P = −∇a∇a + V (x), for V ∈ C∞(M,R) a Klein-Gordon operatoron (M, g). More generally P can be any formally selfadjoint second order differentialoperator, whose principal symbol σpr(P ) equals ξ · g−1(x)ξ.

We denote by Gret/adv the retarded/advanced inverses for P and by G ··= Gret−Gadv, the Pauli-Jordan commutator. We set

(7.4) (u|v)M ··=ˆM

uv dvolg, u, v ∈ C∞0 (M).

The classical phase space associated to P is (V, Q), where

(7.5) V ··=C∞0 (M)

PC∞0 (M), [u] ·Q[v] ··= i(u|Gv)M .

Let Σ be a Cauchy hypersurface,

ρΣu ··=(

uΣi−1∂nuΣ

),

where n is the future unit normal to Σ and VΣ = C∞0 (Σ;C2). We equip VΣ withthe scalar product

(7.6) (f |g)Σ ··=ˆ

Σ

f0g0 + f1g1dσΣ,

Then ρΣG : V → VΣ is bijective, it makes thus sense to define GΣ : VΣ =C∞0 (Σ;C2)→ C∞(Σ;C2) by the identity:

G =·· (ρΣG)∗GΣρΣG,

where the adjoint is taken with respect to the scalar products (7.4), (7.6). Finallywe set

(7.7) fqΣg ··= i(f |GΣg)Σ,

so that the map:ρΣG : (V, Q)→ (VΣ, qΣ)

is pseudo-unitary. One can use equivalently either of the above phase spaces. By acomputation that uses Stoke’s theorem, one has concretely (see e.g. [DG])

(7.8) qΣ =

(0 11 0

).

By the definition of GΣ,

(7.9) 1l = G∗ρ∗ΣGΣρΣ on GC∞0 (M).

This also implies ρΣ = ρΣG∗ρ∗ΣGΣρΣ on GC∞0 (M). On the other hand, denoting

C∞sc (M) the space of space-compact smooth functions (i.e. smooth functions whoserestriction to Σ have compact support), it is well-known that GC∞0 (M) is exactlyKerP |C∞sc (M), see e.g. [BGP]. Furthermore, since the Cauchy problem

(7.10)

Pu = 0,

ρΣu = f.

is well-posed in u ∈ C∞sc (M) for any f ∈ C∞0 (Σ;C2), the map ρΣ : KerP |C∞sc (M) →C∞0 (Σ;C2) is bijective and therefore

(7.11) 1l = ρΣG∗ρ∗ΣGΣ on C∞0 (Σ;C2).


7.1.3. Cauchy evolution operator. It is useful to introduce the Cauchy evolutionoperator:

(7.12) UΣ ··= G∗ρ∗ΣGΣ.

By (7.9) and (7.11), it satisfies ρΣUΣ = 1l on C∞0 (Σ;C2) and UΣρΣ = 1l onKerP |C∞sc (M). Moreover, since G∗ = −G we get PUΣ = 0 hence for f ∈ C∞0 (Σ;C2),u = UΣf is the unique solution in C∞sc (M) of the Cauchy problem (7.10).

7.1.4. Spacetime two-point functions. We use the phase space defined in (7.5). Letus introduce the assumptions:

(7.13)

i) Λ± : C∞0 (M)→ C∞(M)

ii) Λ± ≥ 0 for (·|·)M on C∞0 (M),

iii) Λ+ − Λ− = iG,

iv) PΛ± = Λ±P = 0.

Note that (7.13) implies that Λ± : E ′(M) → D′(M). Let us set with a slightabuse of notation:

u · Λ±v ··= (u|Λ±v), u, v ∈ C∞0 (M).

If (7.13) hold, then Λ± define a pair of complex pseudo-covariances on the phasespace (V, q) defined in (7.5), hence define a unique quasi-free-state on CCR(V, Q).

Definition 7.2. A pair of maps Λ± : C∞0 (M)→ C∞(M) satisfying (7.13) will becalled a pair of spacetime two-point functions.

7.1.5. Hadamard condition. By the Schwartz kernel theorem, we can also identifyΛ± with a pair of distributions Λ±(x, x′) ∈ D′(M × M), and one is especiallyinterested in the subclass of Hadamard states, subject to a condition on the wavefront set of Λ±(x, x′). We recall that the sets N± were defined in 6.6.1.

Definition 7.3. A pair of two-point functions Λ± satisfying (7.13) is Hadamardif

(Had) WF(Λ±)′ ⊂ N± ×N±.This form of the Hadamard condition is taken from [SV, Ho], see also [Wr]

for a review on the equivalent formulations. The original formulation in terms ofwave front sets is due to Radzikowski [Ra], who showed its equivalence with olderdefinitions [KW].

7.1.6. Cauchy surface two-point functions. We will need a version of two-point func-tions acting on Cauchy data of P instead of test functions on M .

Let us introduce the assumptions:

(7.14)

i) λ±Σ : C∞0 (Σ;C2)→ C∞(Σ;C2),

ii) λ±Σ ≥ 0 for (·|·)Σ,

iii) λ+Σ − λ

−Σ = iGΣ.

Definition 7.4. A pair of maps λ±Σ satisfying (7.14) will be called a pair of Cauchysurface two-point functions.

Proposition 7.5. The maps:

λ±Σ 7→ Λ± ··= (ρΣG)∗λ±Σ(ρΣG),

Λ± 7→ λ±Σ ··= (ρ∗ΣGΣ)∗Λ±(ρ∗ΣGΣ)

are bijective and inverse from one another. Furthermore, λ±Σ are Cauchy surfacetwo-point functions iff Λ± are two-point functions.


Prop. 7.5 is proved in [GW2] in a slightly more general context.

7.2. Reduction to the model case. In this subsection we consider a Klein-Gordon operator P on (M, g) in (3.6) satisfying hypotheses (H), (M) introducedin Subsect. 3.3. We show that the construction of Hadamard states for P can bereduced to the case of a model operator P on I ×Σ as introduced in Subsect. 6.2.

We use the notation in Subsect. 3.3. We equip M = I × Σ with the Lorentzianmetric g = −dt2 + ht(y)dy2. We recall that U = χ(M) is an open neighborhood ofΣ in M . We equip C∞0 (U) and C∞0 (M) with their canonical scalar products (·|·)Mand (·|·)M .

Proposition 7.6. Let us set W : C∞0 (U) 3 u 7→ c(d−1)/2u χ ∈ C∞0 (M). Thenthe following holds:(1) There exists a(t, y, ∂y) satisfying the conditions in Subsect. 6.2 for k0 = h0,

such that

P ··= (W−1)∗PW−1 = ∂2

t + r(t, y)∂t + a(t, y, ∂y).

(2) if G is the Pauli-Jordan commutator for P one has G = WGW ∗.(3) Let Λ± be the spacetime two-point functions of a Hadamard state ω for P on

M . Then there exists a unique Hadamard state ω for P on M with two-pointfunctions Λ± such that

Λ± = WΛ±W ∗.

Moreover ω is pure iff ω is pure.

Proof. Without loss of generality we may assume that χ = Id. Let us first prove(1). We set ht = c2ht so that g = −c2dt2 + htdx

2. We have:

P =− |g|− 12 ∂µ|g|

12 gµν∂ν + V

= c−1|h|− 12 ∂tc

−1|h| 12 ∂t − c−1|h|− 12 ∂ich

ij |h| 12 ∂j + V.

A routine computation shows that:

c−1|h|− 12 ∂tc

−1|h| 12 ∂t

= c−1(∂

2

t + c−1|h|− 12 ∂t(c|h|

12 )∂t + |h|− 1

2 ∂t(|h|12 ∂t ln c)

)c−1

=·· c−1P0(t, y, ∂t)c−1,

c−1|h|− 12 ∂ich

ij |h| 12 ∂j + V

= c−1(|h|− 1

2 ∂ichij |h| 12 ∂jc+ c2V

)c−1

= c−1(|h|− 1

2 ∂ihij |h| 12 ∂j + |h|− 1

2 ∂i|h|12hij∂j ln c+ c2V

)c−1

=·· c−1PΣ(t, y, ∂y)c−1.

Using that |h| 12 = cd|h| 12 , we can rewrite these two operators as follows:

P0(t, y, ∂t) = ∂2

t + (|h|− 12 ∂t|h|

12 + (d+ 1)∂t ln c)∂t

+ |h|− 12 ∂t|h|

12 ∂t ln c+ d(∂t ln c)2 + ∂2

t ln c,

PΣ(t, y, ∂y) = |h|− 12 ∂ih

ij |h| 12 ∂j + d∂i ln chij∂j

+ ∂ihij∂j ln c− ∂i|h|−

12 |h| 12hij∂j ln c+ d∂i ln chij∂j ln c+ c2V.


Let now U be the operator of multiplication by c−(d+1)/2. Since

U−1∂tU = ∂t −1

2(d+ 1)∂t ln c, U−1∂iU = ∂i −

1

2(d+ 1)∂i ln c,

we obtain:cU−1PUc = ∂

2

t + r(t, y)∂t + a(t, y, ∂y).

By hypotheses (H), (M) we have r ∈ C∞b (I; BT00(Σ, h0)), a ∈ C∞b (I; Diff2(Σ, h0)),

with principal symbol:σpr(a)(t, y, η) = η · h−1

t (y)η.

If S : C∞0 (U) 3 u 7→ c(d+3)/2u ∈ C∞0 (M) we check that W ∗ = S−1, hencecU−1PUc = SPW−1 = P . Since P is selfadjoint for the scalar product (·|·)Mit follows using dvolg = |ht|

12 dtdy that

r = |ht|−12 ∂t|ht|

12 , a(t, y, ∂y) = a∗(t, y, ∂y).

This completes the proof of (1). From the uniqueness of retarded/advanced inverseswe obtain that Gret/adv = WGret/advW

∗, which proves (2).To prove (3) we use two well-known arguments: the first one is the time-slice

property, which means that V =C∞0 (U)PC∞(U) , i.e. we can replace M by U in (7.5),

since U is a neighborhood of a Cauchy surface. In other words, a pair of two-point functions Λ± ∈ D′(U × U) satisfying (7.13) over U × U uniquely extends toΛ± ∈ D′(M ×M) satisfying (7.13) over M ×M .

The second follows from a result based on Hörmander’s propagation of singular-ities theorem, see [Ra, SV]: if Λ± satisfy (Had) over U ×U , then they satisfy (Had)globally, using that PΛ± = Λ±P = 0. The proof is complete. 2

7.3. The pure Hadamard state associated to a microlocal decomposition.In this subsection we consider the model Klein-Gordon operator P obtained fromProp. 7.6. To simplify notation, we denote P by P , M = I × Σ by M . We willassociate to a microlocal decomposition for P a unique pure Hadamard state. Firstwe need to introduce some more notation.

The level sets Σt = t × Σ are all Cauchy hypersurfaces. The various objectsassociated to the Cauchy surface Σt, like ρΣt , GΣt , λ

±Σt, UΣt will be simply denoted

by ρ(t), G(t), λ±(t), U(t), etc.

Lemma 7.7. One has:i) λ±ω (t) = UA(s, t)∗λ±ω (s)UA(s, t),

ii) c±ω (t) = UA(t, s)c±ω (s)UA(s, t),

iii) q = UA(s, t)∗qUA(s, t),

iv) UA(t, s) = ρ(t)G∗ρ(s)∗G(s).

Proof. It suffices to use the various identities in 7.1.2, 7.1.3 and the fact thatUA(t, s) = ρ(t)U(s). 2

Theorem 7.8. Let U±A (·, ·)(t,s)∈I2 be a microlocal decomposition of the evolutionUA as in Subsect. 6.5 and let λ±(t) ··= ±q c±(t), where c±(t) are defined in Prop.6.6. Then λ±(t) are the Cauchy surface two-point functions of a pure Hadamardstate. One has:

(7.15)

λ±(t) = UA(0, t)∗T−1(0)∗π±T−1(0)UA(0, t)

= T−1(t)∗UC(0, t)∗π±UC(0, t)T−1(t)

= T−1(t)∗π±T−1(t) + C∞b (I; Ψ−∞(Σ)).


where π± are defined in (6.25) and T (t) in (6.15).

Proof. Let us first prove (7.15). From the definition of c±(t) (see (6.26), (6.29)),we have:

λ±(t) = ±qUA(t, 0)T (0)π±T−1(0)UA(0, t)

= ±UA(0, t)∗qT (0)π±T−1(0)UA(0, t)

= ±UA(0, t)∗T−1(0)∗qπ±T−1(0)UA(0, t)

= UA(0, t)∗T−1(0)∗π±T−1(0)UA(0, t),

where we used successively (6.8), (6.16) and the fact that ±qπ± = π±. The secondline in (7.15) follows then from (6.14). From (6.17), (6.22) we obtain then

q = UC(0, t)∗qUC(0, t) = UC(0, t)∗qUC(0, t) + C∞b (I,Ψ−∞(Σ)).

Since q and UC(0, t) are diagonal this implies that

π± = UC(0, t)∗π±UC(0, t) + C∞b (I,Ψ−∞(Σ)),

which using once more (6.22) gives:

π± = UC(0, t)∗π±UC(0, t) + C∞b (I,Ψ−∞(Σ)),

which completes the proof of (7.15).To check that λ±(t) are the Cauchy surface two-point functions of a pure Ha-

damard state we work with the Cauchy surface Σ0. By Prop. 6.6 we know thatc+(0) + c−(0) = 1l hence condition (7.14) iii) is satisfied. Condition i) follows fromthe fact that c±(0) ∈ Ψ∞(Σ). The positivity condition iii) follows from (7.15).To check the Hadamard condition one can use the arguments in [GW2], which werecall for the sake of self-containedness. We have by (6.28):

U(0)c±(0) = U±A (·, 0)c±(0) on E ′(Σ;C2),

henceΛ± = ±iU(0)c±(0)ρ(0)G = ±iU±A (·, 0)ρ(0)G, on E ′(M).

From Thm. 6.5 this implies that WF′(Λ±) ⊂ N±×N . Since Λ± = (Λ±)∗ we havealso WF′(Λ±) ⊂ N± × N±. This shows that the state ω is Hadamard. The factthat ω is pure follows from the fact that c±(0) are projections. 2

We recall from Subsects. 6.4.3, (6.5) that a microlocal decomposition U±A (·, ·) isuniquely obtained from a generator b(t) constructed in Thm. 6.1 as an approximatesolution of a Riccati equation. Consequently to any such b(t) corresponds a uniquepure Hadamard state.

Definition 7.9. The pure Hadamard state associated to a generator b(t) in Thm.7.8 will be denoted by ωb.

It is now easy to find the relationship between the spacetime two-point functionsΛ± of ω and the operators UA(t, s). As in Subsect. 6.6 we denote by A(t, s) thetime kernel of an operator A.

Theorem 7.10. Let Λ± be the spacetime two-point functions of the state ω con-structed in Thm. 7.8. Then:

(7.16) U±A (t, s) = ±(

i∂sΛ±(t, s) Λ±(t, s)

∂t∂sΛ±(t, s) i−1∂tΛ

±(t, s)

).

Consequently we have:

(7.17) Λ±(t, s) = ±π0U±A (t, s)π∗1 ,

where πi are defined in Subsect. 6.6.


Proof. Using (6.28) and the identities in Lemma 7.7, Prop. 7.5 we obtain:

U±A (t, s) = UA(t, 0)c±(0)UA(0, s)

= ρ(t)G∗ρ(0)∗G(0)c±(0)ρ(0)G∗ρ(s)∗G(s)

= ±iρ(t)G∗ρ(0)∗λ±(0)ρ(0)G∗ρ(s)∗G(s)

= ±iρ(t)Λ±ρ(s)∗G(s) = ±ρ(t)Λ±ρ(s)∗q.

Using ρ(s)∗f = f0 ⊗ δ(s)− if1 ⊗ δ′(s), this yields:

U±A (t, s) = ±(

i∂sΛ±(t, s) Λ±(t, s)

∂t∂sΛ±(t, s) i−1∂tΛ

±(t, s)

),

which completes the proof. 2

In Subsect. 6.6 we associated to a microlocal decomposition a canonical Feynmaninverse GF, see Thm. 6.8. On the other hand, it is well-known (see e.g. [Ra] or [Wr,Thm. 3.4.4] for the complex case) that if Λ± are the spacetime two-point functionsof a Hadamard state ω, then the operator

i−1Λ+ +Gadv = i−1Λ− +Gret

is a Feynman inverse of P . It is easy to show that if ω is the state in Thm. 7.8then both Feynman inverses are the same.

Proposition 7.11. Let GF and Λ± the Feynman inverse and spacetime two-pointfunctions associated to the microlocal decomposition U±A (·, ·)(t,s)∈I2 . Then

GF = i−1Λ+ +Gadv = i−1Λ− +Gret

Proof. Arguing as in the proof of Thm. 6.8 we see that

Gret(t, s) = i−1π0UA(t, s)π∗1θ(t− s), Gadv(t, s) = −i−1π0UA(t, s)π∗1θ(s− t).Then the proposition follows from the identities (6.31), (7.17). 2

7.4. Regular Hadamard states.

Definition 7.12. A state ω is regular if λ±ω (t) ∈ C∞(R,Ψ∞(Σ;M2(C))).

In other words regular states have Cauchy surface two-point functions equal tomatrices of pseudodifferential operators. The following lemma shows that it sufficesto check the pseudodifferential property for one time t.

Lemma 7.13. ω is regular iff there exists s ∈ I such that λ±ω (s) ∈ Ψ∞(Σ;M2(C)).

Proof. Assume that λ±ω (s) ∈ Ψ∞(Σ;M2(C)) for some s ∈ I. Then λ±ω (t) is givenby Lemma 7.7 i). By Thm. 6.3 we can replace UA(s, t) by UA(s, t) and thenby UC(s, t), which has a diagonal generator, see (6.19). Then we apply Egorov’sTheorem, Thm. 5.15. 2

Let now ω be the Hadamard state associated to a microlocal decomposition asin Thm. 7.8, and ω1 another regular state. We denote by Λ±, Λ±1 , λ

±(t), λ±1 (t)their respective spacetime and Cauchy surface two-point functions.

Proposition 7.14. A regular state ω1 is Hadamard iff:

(7.18) λ±1 (t)− λ±(t) ∈ C∞(R; Ψ−∞(Σ;M2(C))).

Proof. It is well known (see e.g. [Ra] or [Wr, Thm. 3.4.4] for the complex case)that Λ± − Λ±1 are smoothing operators on M , hence λ±(t)− λ±1 (t) are smoothingoperators on Σ. By Lemma 5.13 this yields the ⇒ implication. The ⇐ implicationis immediate. 2


7.4.1. Bogoliubov transformations. We work in the setup of 7.1.1. It is well known,see e.g. [DG, Thm. 11.20], that if ω, ω1 are two pure quasi-free states on CCR(V, q),then there exists u ∈ U(V, q) such that

λ±1 = u∗λ±u.

We now examine the form of the operator u if ω is a pure Hadamard state associatedto a microlocal decomposition as in Thm. 7.8 and ω1 is another pure, regularHadamard state. In the proposition below, we fix the reference time t = 0. Theoperator b(0) ∈ Ψ1(Σ) entering in the definition of λ±(0) (see formulas (7.15) and(6.26)) will be simply denoted by b.

Proposition 7.15. Let λ±1 (0) be the t = 0 two-point functions of a pure regularHadamard state ω1. Then there exists a ∈ Ψ−∞(Σ) such that:

(7.19) λ+1 = T−1(0)∗U∗π+UT−1(0), for U =

((1l + aa∗)

12 a

a∗ (1l + a∗a)12

).

Proof. Let us set

η1 = λ+1 + λ−1 , η1 = T−1(0)∗η1T

−1(0).

Since ω1 is pure, we deduce from Prop. 7.1 and identity (6.16) that η1 satisfies:

(7.20) i) η1 ≥ 0, ii) η1qη1 = q,

where we recall that q =

(1 00 −1

). We write η1 as

(a bb∗ c

), where using

(7.18) and the fact that η = 1l we know that b, 1l− a, 1l− c ∈ Ψ−∞(Σ). Now (7.20)is equivalent to:

i′) a ≥ 0, c ≥ 0, |(u|bv)| ≤ (u|au)12 (v|cv)

12 , u, v ∈ C∞0 (Σ),

ii′) a2 = 1l + bb∗, c2 = 1l + b∗b, ab− bc = 0.

Since a, c ≥ 0 by i’), the first two equations of ii’) yield

a = (1l + bb∗)12 , c = (1l + b∗b)

12 .

The third equation of ii’) then holds using the identity

(7.21) bf(b∗b) = f(bb∗)b, f any Borel function.

The second condition in i’) is equivalent to ‖(1l + bb∗)12 b(1l + b∗b)

12 ‖ ≤ 1, which

holds using again (7.21). This implies that λ1 ··= T−1(0)∗λ+1 T−1(0) equals

λ1 =1

2

((1l + bb∗)

12 + 1l b

b∗ (1l + b∗b)12 − 1l

).

Let nowa ··=

b√2

((1l + b∗b)12 + 1l)

12 ∈ Ψ−∞(Σ).

Using (7.21) we obtain by an easy computation that

1l + a∗a =1

2((1l + b∗b)

12 + 1l), 1l + aa∗ =

1

2((1l + bb∗)

12 + 1l), b = 2a(1l + a∗a)

12 .

Hence

λ1 =

(1l + aa∗ a(1l + a∗a)

12

(1l + a∗a)12 a∗ a∗a

)= U∗π+U,

for U as in the proposition. 2

The following theorem shows that any pure regular Hadamard state is actuallyassociated to a microlocal decomposition.


Theorem 7.16. Let ω1 be a pure regular Hadamard state. Then there exists agenerator b1(t) as in Thm. 6.1 such that ω1 = ωb1 .

Before proving Thm. 7.16 we need one more lemma.

Lemma 7.17. Let a ∈ Ψ−∞(Σ) and set r(a) ··= (1l + aa∗)12 − a. Then r(a) is

boundedly invertible with r(a)−1 ∈ 1l + Ψ−∞(Σ).

Proof. By the polar decomposition theorem, we have a = u|a| = |a∗|u, where uis a partial isometry. Moreover, r(a) = (1l + aa∗)

12 (1l − (1l + aa∗)−

12 a). To prove

invertibility it suffices to notice that (1l + aa∗)−12 a = (1l + |a∗|2)−

12 |a∗|u has norm

< 1, which is easily checked by using the self-adjoint functional calculus and the factthat a∗ is bounded. The fact that r(a)−1 − 1l ∈ Ψ−∞(Σ) follows by the argumentused already to prove Lemma 5.5. 2

Proof of Thm. 7.16. From Prop. 7.15, we know that there exists a ∈ Ψ−∞(Σ)such that (7.19) holds. Let us first try to find some b1 ∈ Ψ1(Σ) such that

(7.22) λ+1 = (T−1

1 )∗π+T−11 ,

where T1 is defined as in (6.15) with b = b(0) replaced by b1. The proof is dividedin several steps.

Step 1: we first solve (7.22). Let r(a) be as in Lemma 7.17 and set

(7.23) z ··= r(a)(b+ b∗)−12 ∈ Ψ−

12 (Σ).

Note that z−1 ∈ Ψ12 (Σ). We claim that

(7.24) b1 ··= b+ (b+ b∗)12 a∗z∗−1 = b+ Ψ−∞(Σ)

solves (7.22). Indeed, if V1 =

(α1 β1

γ1 δ1

)the equation

V ∗π+V = V ∗1 π+V1,

is equivalent to the system

(7.25)

i) α∗1α1 = α∗α

ii) α∗1β1 = α∗β,

iii) β∗1β1 = β∗β.

If V = UT (b) and V1 = T (b1) we have:

α1 = (b1 + b∗1)−12 b∗1, β1 = (b1 + b∗1)−

12 ,

α = (1 + aa∗)12 (b+ b∗)−

12 b∗ + a(b+ b∗)−

12 b,

β = (1 + aa∗)12 (b+ b∗)−

12 − a(b+ b∗)−

12 .

Using the operator z introduced in (7.23) we see that

(7.26) α = zb∗ + a(b+ b∗)12 , β = z.

Note also that:

(7.27)r(a)r(a)∗ + r(a)a∗ + ar(a)∗

= (r(a) + a)(r(a)∗ + a∗)− aa∗ = 1.


Hence for b1 given by (7.24) we have:

b1 + b∗1 = (b+ b∗) + (b+ b∗)12 a∗z∗−1 + z−1a(b+ b∗)−1

= (b+ b∗)12

(1 + a∗r(a)∗−1 + r(a)−1a

)(b+ b∗)

12

= z−1(r(a)r(a)∗ + z(b+ b∗)

12 a∗ + a(b+ b∗)

12 z∗)z∗−1

= z−1 (r(a)r(a)∗ + r(a)a∗ + ar(a)∗) z∗−1 = z−1z∗−1,

by (7.27), hence:β∗1β1 = (b1 + b∗1)−1 = z∗z = β∗β,

hence (7.25) iii) is satisfied. We also obtain

α∗1β1 = b1(b1 + b∗1)−1 = b1z∗z

= bz∗z + (b+ b∗)12 a∗z = α∗β,

hence (7.25) ii) is satisfied. Finally we have

α∗1α1 = b1(b1 + b∗1)−1b∗1 = b1z∗zb∗1

= (bz∗ + (b+ b∗)12 a∗)(zb∗ + a(b+ b∗)

12 ) = α∗α,

by (7.26). Therefore (7.25) i) is satisfied and b1 solves (7.22).Step 2: we check that b1 = b1(0) satisfies the properties i), ii) and iii) in Thm.

6.1 at t = 0. First of all b1 = b+ Ψ−∞(Σ) hence i) is satisfied. We claim that

(7.28) b1 + b∗1 ∼ b+ b∗,

(see Subsect. 1.4 for notation), which implies properties ii), iii) at t = 0. In factwe have:

b1 + b∗1 = (b+ b∗)12

(1 + a∗r(a)−1∗ + r(a)−1a

)(b+ b∗)

12

= (b+ b∗)12 r(a)−1 (r(a)r(a)∗ + r(a)a∗ + ar(a)∗) r(a)−1∗(b+ b∗)

12

= (b+ b∗)12 r(a)−1 ((r(a) + a)(r(a)∗ + a∗)− aa∗) r(a)−1∗(b+ b∗)

12

= (b+ b∗)12 r(a)−1r(a)−1∗(b+ b∗)

12 ,

which implies (7.28) since r(a) is boundedly invertible by Lemma 7.17.Step 3: we now extend b1 into b1(t). We set

b1(t) = b(t) + r−∞(t),

where r−∞ ∈ C∞(R,Ψ−∞(Σ)) is chosen such that r−∞(0) = b1(0) − b(0) andproperties i), ii), iii) are satisfied for all t ∈ I. Then iv) is automatically satisfiedalso. This completes the proof of the theorem. 2

Appendix A

A.1. Computations for Kerr-de Sitter. We recall now an identity of the kindwhich is often used in the literature (see e.g. [O’N2, Lemma 2.2.1] for the Kerrmetric).

Remark A.1. The identity in Lemma A.2 allows to check that the Kerr-de Sittermetrics are smooth Lorentzian metrics on M despite the fact the forms dϕ, dθ aresingular at the poles of S2. In fact the forms sin 2θdθ and sin2 θdϕ are smooth onS2, since they equal xdx + ydy and xdy − ydx in Cartesian coordinates near thepoles, and the standard metric on S2 equals dθ2 + sin2 θdϕ2.


Lemma A.2. Let

dω2 = dθ2 +1 + α cos2 θ

1 + αsin2 θdϕ2

Then:(1) dω2 is a smooth Riemannian metric on S2.(2) One has:

gθθdθ2 + gϕϕdϕ

2 =ρ2

∆θdω2 +

(2Ma2r

(1 + α)2ρ2+

a2

1 + α

)(sin2 θdϕ)2.

Proof. dω2 is clearly positive definite. We have

dω2 = (dθ2 + sin2 θdϕ2)− α

1 + α(sin2 θdϕ)2.

The first term is the standard metric on S2, the second term is smooth since sin2 θdϕis a smooth 1−form on S2. Therefore dω2 is smooth, which proves (1).

A routine computation shows that:

σ2 = (r2 + a2)(1 + α)ρ2 + 2Ma2r sin2 θ.

Using this identity in the definition of gϕϕ (see (4.2)) we easily obtain (2). 2

A.1.1. Some classes of functions. The map I 3 r 7→ s(r) ∈ R is bijective. Setting:

(KdS) κh/c ··= ∓∂r∆r(rh/c)

(1+α)(r2h/c

+a2),

(K) κh ··= ∓ ∂r∆r(rh)(1+α)(r2h+a2)

,

which are related to the surface gravities at the Kerr-de Sitter resp. Kerr case, onehas:

(KdS) (r − rh/c) ∼ e−κh/c|s|, ∂αs (r − rh/c) ∈ O(e−κh/c|s|), for s→ ∓∞,

(K)

(r − rh) ∼ e−κh|s|, ∂αs (r − rh) ∈ O(e−κh|s|), for s→ −∞,r ∼ s, ∂αs r ∈ O(〈s〉1−α), for s→ +∞.

Definition A.3. We set:T 0

KdS = f ∈ C∞(]rh, rc[×S2) : ∂αr ∂βωf ∈ O(1),

T 0,0K = f ∈ C∞(]rh,+∞[) : ∂αr ∂

βωf ∈ O(〈r〉−α)

T pKdS = (r − rh)p(r − rc)pT 0KdS, p ∈ Z,

Tm,pK = 〈r〉m−p(r − rh)pT 0,0K ,m ∈ R, p ∈ Z.

The following are the images of the above spaces under the change of variabler 7→ s(r).

Definition A.4. We set:SpKdS = f ∈ C∞(R× S2) : ∂αs ∂

βωf ∈ O(epκh/c|s|), ±s < 0, p ∈ Z∗,

S0KdS = f ∈ C∞(R× S2) : f bounded , ∂sf ∈ S−1

KdS,

Sm,pK = f ∈ C∞(R× S2) : ∂αs ∂βωf ∈ O(epκh|s|), s < 0, ∂αs ∂

βωf ∈ O(〈s〉m−α), s > 0, p ∈ Z∗,

Sm,0K = f ∈ C∞(R× S2) : ∂βωf ∈ O(〈s〉m), ∂sf ∈ Sm−1,−1K .

Definition A.5. A function f ∈ SpKdS, resp. f ∈ Sm,pK is elliptic if f(s, ω) 6= 0 on

R× S2 and f−1 ∈ S−pKdS, resp. f−1 ∈ S−m,−pK .

The following result is easy to prove (see [Hä, Sect. 9.3]).


Lemma A.6. (1) Sp1KdS × Sp2KdS ⊂ S

p1+p2KdS and Sm1,p1

K × Sm2,p2K ⊂ Sm1+m2,p1+p2

K ,(2) Set f(s, ω) = f(r(s), ω) for f ∈ C∞(I×S2). Then if f ∈ T pKdS, resp. f ∈ T

m,pK

one has f ∈ S−pKdS, resp. f ∈ Sm,−pK .

From Lemma A.6 we obtain easily the following lemma.

Lemma A.7. One has:

i) ρ2, r2 + a2 are elliptic in S0KdS, resp. in S2,0

K ,

ii) σ2 is elliptic in S0KdS, resp. in S4,0

K ,

iii) ∆θ is elliptic in S0KdS, resp. in S0,0

K ,

iv) ∆r is elliptic in S−1KdS, resp. in S2,1

K

v) F (s) ··= (1 + α)2 (r2+a2)2

∆ris elliptic in S1

KdS, resp. in S2,1K ,

vi) G(s, θ) ··= σ2

(r2+a2)2∆θ, is elliptic in S0

KdS, resp. in S0,0K ,

vii) c2 = ∆r∆θρ2

(1+α)2σ2 ∈ S−1KdS resp. ∈ S0,−1

K .

Proof. Statements i), . . . , iv) are routine computations, using Lemma A.6 (2).The remaining statements follow then from Lemma A.6 (1). 2

In the next lemma we estimate the function R defined in 4.2.2.

Lemma A.8. Let R = gtϕg−1ϕϕ and set

Rr(s, θ) = ∂rR(r(s), θ), Rθ(s, θ) ··= (sin 2θ)−1∂θR(r(s, θ)).

Then:

(A.1)Rr ∈ S0

KdS, resp. ∈ S0,−3K ,

Rθ ∈ S−1KdS, resp. ∈ S−1,−2

K .

Proof. We have

R(r, θ) = − aσ2

(∆r − (r2 + a2)∆θ

)=

ar2 + a2

(1− ∆r

(r2 + a2)∆θ

)(1− a2∆r sin2 θ

(r2 + a2)2∆θ

)−1

,

henceR =

ar2 + a2

(1 +R1(r, θ)), R1 ∈ T 1KdS, resp. R1 ∈ T 1,−1

K .

It follows that (using that R depends on θ only through sin2 θ):

Rr ··= ∂rR ∈ T 0Kds, resp. ∈ T

0,−3K ,

Rθ ··= (sin 2θ)−1∂θR ∈ T 1KdS, resp. ∈ T

1,−2K .

Passing to the variable s using Lemma A.6 we obtain (A.1). 2

A.2. Proof of Prop. 7.1. Let us recall that in our notations, Vcpl is the com-pletion of V with respect to the Hilbert norm ‖v‖2ω ··= vΛ+v + vΛ−v, and thesuperscript ‘cpl’ is also used to denote canonical extensions of various objects on Vto Vcpl.

If V = Vcpl, then the real symplectic space (V,Reσ) is complete for the Euclideannorm (vηv)

12 , η = Re(Λ± ∓ 1

2q). In that situation, we know from [DG, Thm.17.13] that ω is pure iff (2ηcpl,Reσcpl) is Kähler, i.e. there exists an anti-involution


j1 ∈ Sp(Vcpl,Reσcpl) such that 2ηcpl = Reσcplj1. This is known to be equivalentto the existence of projections c± satisfying

(A.2) c+ + c− = 1l, c+∗qc− = 0, Λ± = ±q c±,

as requested, see [GW1, Prop. 2.7].Let us now treat the general case. We recall from [BR, Thm. 2.3.19] that a state

ω on a C∗-algebra A is pure iff its GNS representation (Hω, πω) is irreducible, i.e.iff Hω does not contain non-trivial closed subspaces invariant under πω(A).

For V1 as in the statement of the proposition, we set A(1) = W(V(1), q(1)), andwe let (H(1), π(1),Ω(1)) be the GNS triple for (A(1), ω(1)). Using that V is dense inV1 for ‖·‖ω, we first obtain that H = H1, Ω = Ω1 and π1|A = π.

We also easily obtain that π(A) is strongly dense in A1. In fact, if A =∑N1 λiπ1(W (vi)) ∈ π1(A1) and vi,n ∈ V with vi,n → vi for ‖·‖ω, we obtain that

An =∑N

1 λiπ(W (vi,n)) is bounded by∑N

1 |λi| and converges strongly to A on thedense subspace π(A)Ω, hence on H.

From this fact we see that a closed subspace K ⊂ H is invariant under π(A) iffit is invariant under π1(A1), hence ω is pure iff its extension ω1 to A1 is pure.

Therefore, the ⇒ direction is shown simply by taking V1 = Vcpl. Conversely,if on a space V1 as in the statement of the proposition, there exist projections c±1satisfying (A.2) (with c±1 ,Λ

±1 in place of c±,Λ±), then an easy computation shows

that as identities on L(V1,V∗1 ), one has

c±∗1 λ±1 c±1 = λ±1 , c±∗1 λ∓1 c

±1 = 0,

hence c±1 are bounded for ‖ · ‖ω. Therefore they extend to projections on Vcpl

satisfying (A.2). This implies that ωcpl is pure, hence ω is pure. 2

Acknowledgments. The authors would like to thank Victor Nistor and Jean-Philippe Nicolas for useful conversations. M.W. gratefully acknowledges the France-Stanford Center for Interdisciplinary Studies for financial support and the Depart-ment of Mathematics of Stanford University for its kind hospitality. The authorsalso wish to thank the Erwin Schrödinger Institute in Vienna for its hospitalityduring the program “Modern theory of wave equations”.

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University of Göttingen 2013.

Département de Mathématiques, Université Paris-Sud XI, 91405 Orsay Cedex,France

E-mail address: [email protected]

Département de Mathématiques, Université Paris-Sud XI, 91405 Orsay Cedex,France


Université Grenoble Alpes, Institut Fourier, UMR 5582 CNRS, CS 40700, 38058Grenoble Cedex 09, France


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